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Melting pot or salad bowl: the formation of heterogeneous communities
IFS Working Paper W15/30
Arun Advani Bryony Reich
The Institute for Fiscal Studies (IFS) is an independent research institute whose remit is to carry out rigorous economic research into public policy and to disseminate the findings of this research. IFS receives generous support from the Economic and Social Research Council, in particular via the ESRC Centre for the Microeconomic Analysis of Public Policy (CPP). The content of our working papers is the work of their authors and does not necessarily represent the views of IFS research staff or affiliates.
Melting Pot or Salad Bowl: The Formation of Heterogeneous
Communities
Arun Advani
Institute for Fiscal Studies
and University College London
Bryony Reich∗
University College London
October 2015
Abstract
Relatively little is known about what determines whether a heterogenous population ends up in a coop-
erative or divisive situation. This paper proposes a theoretical model to understand what social structures
arise in heterogeneous populations. Individuals face a trade-off between cultural and economic incentives:
an individual prefers to maintain his cultural practices, but doing so can inhibit interaction and economic
exchange with those who adopt different practices. We find that a small minority group will adopt majority
cultural practices and integrate. In contrast, minority groups above a certain critical mass, may retain diverse
practices and may also segregate from the majority. The size of this critical mass depends on the cultural
distance between groups, the importance of culture in day to day life, and the costs of forming a social tie.
We test these predictions using data on migrants to the United States in the era of mass migration, and
find support for the existence of a critical mass of migrants above which social structure in heterogeneous
populations changes discretely towards cultural distinction and segregation.
1 Introduction
Ethnic, linguistic, and religious heterogeneity are associated with a variety of politico-economic
problems, including low growth, low provision of public goods, and conflict. Yet the effects of
diversity are not uniform: some heterogenous populations manage to avoid societal conflict and
are economically successful. In some cases, diversity can even have positive effects on growth,
productivity, and innovation.1 Seeking to explain this puzzle, recent work finds that social
cohesion plays an important role in how well a population ‘deals’ with its heterogeneity. For
example, more segregated populations are found to have lower quality of governance, lower
trust between citizens, and worse education and employment outcomes, compared to similarly
heterogenous populations that are better integrated.2
∗Corresponding author: Faculty of Economics, University College London. Email: [email protected]
We are extremely grateful to Alberto Alesina, Sanjeev Goyal, and Imran Rasul. For very helpful comments we would like to thank
Ana Babus, Tim Besley, Antonio Cabrales, Partha Dasgupta, Christian Dustmann, Andrea Galeotti, Raffaella Giacomini, Sriya Iyer,
Matthew Jackson, Terri Kneeland, Edward Lazear, Friederike Mengel, Paul Milgrom, Jonathan Newton, Marco van der Leij, Chris
Parmeter, Ilya Segal, Jorgen Weibull, and Peyton Young, as well as audiences at ASREC Washington DC, Barcelona Summer
Forum, Cambridge, CTN Warwick, Oxford, PET Lisbon, SAET Paris, and Stanford. Advani gratefully acknowledges financial
support from the ERC (GA313234). Reich gratefully acknowledges financial support from the NET Institute (www.NETinst.org)
and the UK Economic and Social Research Council (grant number ES/K001396/1).1Easterly and Levine (1997), Alesina et al. (1999), Montalvo and Reynal-Querol (2005), Ashraf and Galor (2013), and Rasul
and Rogger (2015), amongst others.2See for example Alesina and Zhuravskaya (2011) and Cutler and Glaeser (1997). Firm and team studies suggest that environ-
ments that bring together heterogeneous skills and ideas, at the same time as fostering cooperation, are associated with positive
effects of diversity on innovation and productivity. At the national level Ashraf and Galor (2013) show a hump-shaped relationship
1
Despite its clear importance, relatively little is known about what determines whether a het-
erogenous population ends up in a cooperative or divisive situation. In this paper we propose
a model that allows us to study this question. We ask, first, what social structures can arise
when society is composed of heterogeneous groups, and second, what features influence which
social structure arises in a particular environment. Understanding this is crucial in an increas-
ingly mobile world in which immigration and social cohesion are frequently at the forefront of
political agendas.3
In our model, individuals from two distinct groups – natives and immigrants – living side-
by-side, make choices over activities to engage in, and who in the community to interact with.
Some activities, such as language choice, are ‘cultural actions’, in the sense that each group will
start with an ex ante preferred language, and there is a cost to switching to any other choice.
Other activities are ‘non-cultural’: there is no group-specific reason why one activity is more
costly to engage in than another.4 For example, there may be no group-specific reason why one
sport should be more costly to engage in than another.5 Interaction provides opportunities for
economic exchange, and so individuals can benefit from forming social ties with others in the
population. Since interaction requires some degree of commonality of actions, we assume that
the benefit of a link is increasing in the number of shared activities, with a fixed (per-link) cost
of formation. The fundamental trade-off is that an individual prefers to maintain his cultural
practices but doing so can hinder opportunities for interaction and exchange with those who
adopt different practices.6
We find that only three classes of social structure are possible in (Nash) equilibrium: as-
similation, segregation, and multiculturalism. In assimilation equilibria, all individuals engage
in the same activities (cultural and non-cultural), and interact with everyone. In segregation
equilibria, all individuals play their type-specific cultural activities, and form social ties only
with those of the same type. In multicultural equilibria, individuals play their type-specific
cultural activities, but all coordinate on common non-cultural activities, and this allows indi-
viduals to form ties with all others, regardless of type. The existence of shared non-cultural
practices allows interaction between disparate groups who maintain distinct cultures.
What kind of environments sustain these three outcomes? The key pattern we identify
is that social structure in heterogeneous populations changes discretely in the share of the
minority. When the minority (immigrant) group is small relative to the majority, assimilation
occurs. Intuitively, restricting social interaction within such a small group is not desirable and
between genetic diversity and growth, consistent with their hypothesis that diversity is beneficial for innovation and production
but can also be costly when it reduces cooperation and increases disarray within a society. See Alesina and La Ferrara (2005) and
Alesina et al. (2013) for further discussion and references.3Europe provides an example of a variety of policies designed to integrate immigrants and perhaps also influence voters. For
example, in France, in 2010, the senate voted 246 − 1 to ban the full Islamic veil, with the French President arguing it was not
consistent with French identity. In the UK, in 2013, the government announced plans to increase English language requirements to
ensure migrants were ‘able to integrate into British society’.4In the model presented in the paper an individual chooses only two actions, a cultural action and a non-cultural action. In
Online Appendix B we allow for multiple cultural and non-cultural actions.5Of course some groups have ties to particular sports and so which activities are non-cultural and cultural will depend on the
groups. A word of caution: a result of this paper is that non-cultural activities can in equilibrium become associated with a
particular group. Thus many of the examples that spring to mind that associate sport or other activities with a particular group
may in fact be an equilibrium outcome and not a result of some ex ante preferences.6In an example of this, Algan et al. (2013) find an economically significant trade-off faced by Arabic parents in France between
attachment to their own culture (in their study, the desire to pass on an Arabic name) with the future economic performance of
their children in the form of work-related penalties to having an Arabic name. This is an important trade-off and is present in
varying forms in Kuran and Sandholm (2008), Lazear (1999), Bisin et al. (2011), Carvalho (2013b), and Carvalho (2013a), amongst
others.
2
an immigrant does better by paying the cost of switching culture and being absorbed into a
much larger group. Multiculturalism and segregation are equilibria only once the share of the
minority group in the population exceeds particular (different) thresholds. Above a certain
threshold, there is a large enough ‘critical mass’ of immigrants that if the group maintains its
distinct culture then, for any immigrant, the cost of switching culture outweighs the benefits
of increased interaction. This threshold result is due to complementarities: maintaining a
distinct minority culture is only ‘worth it’ if others do so, therefore either no one maintains the
immigrant culture or a large group of immigrants does.
The location of these thresholds, above which segregation and multiculturalism are equi-
libria, depends on three parameters: (i) cultural distance; (ii) the importance of culture in
everyday life; and (iii) the cost of forming a social tie. When the cultural distance between
immigrants and natives is larger, the cost of switching cultural actions is relatively higher. This
reduces the threshold immigrant share at which multiculturalism and segregation – equilibria
where immigrants retain their own culture – become possible. In contrast, a higher importance
of culture in daily life makes having a common culture more important for interaction, and so it
is harder to sustain multiculturalism (the threshold on multiculturalism rises). To illustrate, if
social interaction involving alcohol is ubiquitous, and an immigrant’s culture prohibits alcohol,
then this makes it difficult for the immigrant to both maintain his cultural practices and inte-
grate with natives, and thus he is forced to choose. Finally, higher costs of forming a link lower
the threshold on segregation. Intuitively, since the costs of assimilation come from switching
culture, while the benefits come from improved interaction with natives, a higher cost of link
formation lowers the relative benefits of assimilation and makes segregation easier to sustain.
We extend our model to account for multiple generations, where we consider cultural trans-
mission from parents to children.7 Considering transmission allows us to see which equilibria
are most likely in the long run and how robust our static results are. We find the same general
pattern as the static model: the three classes of social structure can exist in the long-run; there
is a distinct threshold for each structure, based on the share of immigrants in the population;
and these thresholds depend on cultural distance, the importance of culture in everyday life,
and the cost of forming a link.
We test these predictions using census data on immigrant populations in the United States
at the beginning of the 20th Century. During this period large numbers of immigrants arrived
in America, radically altering the national make-up of the country. This is exactly the kind of
situation our framework seeks to understand. Indeed, the three distinct forms of social structure
we find are evident in scholarly discussion of communities in the United States following mass
migration (Gordon, 1964). Early scholars argued that a single culture would prevail: the so-
called ‘melting pot’. As it became clear that not all communities assimilated, but some instead
‘retained distinctive economic, polical and cultural patterns long after arriving in the United
States’, segregation became a major concern (Bisin and Verdier, 2000). However, there was also
discussion of a ‘third way’ – ‘the salad bowl’ – where immigrants could become ‘American’ and
integrate whilst maintaining some cultural distinction. This period in history has the added
benefit of furnishing us with data on a large number of heterogeneous communities with varying
immigrant group sizes, which is vital to any chance of observing thresholds.
7Formally we use stochastic stability techniques, see Young (1993) and Kandori et al. (1993). Children inherit their parent’s
culture, and choose how to best respond to the existing actions of the population, but occasionally ‘experiment’ by playing
suboptimal actions.
3
If our framework captures the key forces driving community formation, then we should
expect behavior in heterogeneous populations to exhibit the threshold patterns predicted by
(and central to) the model. We test two key predictions of the model: that a threshold should
exist in community behavior, and that the location of this threshold should depend on cultural
distance from the existing population of the US. These predictions also allow us to separate our
model from explanations based on selection, which do not suggest the discontinuities between
group size and community outcomes that are critical to our model. We test our predictions on
the decision of immigrants to acquire the English language or not (a cultural action), and on the
decision to in-marry (a partial measure of interaction). We find sharp and significant thresholds
in behavior when an immigrant group forms around one-third of the community: above this
threshold English acquisition in the immigrant group falls by almost a half, from 90% to 50%,
and in-marriage rises by a third from 55% to 75%. Using data on linguistic distance between the
language of the immigrant group and English, we additionally show that, as predicted, when
cultural (linguistic) distance is higher, the estimated threshold for segregation and multicultural
outcomes is lower.
The literature examining choice of culture highlights the variety of outcomes that can arise
in heterogenous populations. In seminal work in economics on cultural transmission, Bisin and
Verdier (2000) examine the persistence of different traits in a mixed community even in the long
run.8 Iannaccone (1992), Berman (2000), and Carvalho (2013b) study the stability of costly
and restrictive cultural practices within religious groups in heterogenous societies.9 Kuran and
Sandholm (2008) examine the convergence of cultural practices when diverse groups interact.
The question we ask in this paper is different. We want to understand selection between various
outcomes.10 That is, when does one of these social structures emerge rather than another?
It is important to highlight the two features of the theoretical framework that make it rich
enough to produce the different social structures. The first is the interaction between choice of
behavior and choice of interaction, discussed further below. The second is the novel introduction
of non-cultural actions, which play an important role in uniting or dividing communities. Adop-
tion of a common non-cultural action is necessary for the existence of multicultural equilibria.
At the other end of the spectrum, rather than bridging the gap between groups, non-cultural
actions can also be used to divide them. We find a subclass of segregation equilibria in which
immigrants not only retain distinct cultural practices but also create ‘new diversity’ by adopt-
ing deliberately different non-cultural activities from natives. Such polarization of non-cultural
practices occurs in order to maintain segregation by raising the cost of interacting with the
other group. This extreme form of segregation occurs when culture is relatively unimportant
in everyday life.
Close to our question, in pioneering work, Lazear (1999) studies the choice of whether or not
8See also Bisin et al. (2004) for empirical work on this topic. A key finding in Bisin and Verdier (2000), that smaller groups
may exert more effort in passing on their culture to their children, sounds at odds with our finding that smaller groups are more
likely to assimilate. This is not the case. When groups continue to maintain different practices in equilibrium we find that smaller
groups must put in more ‘effort’ in the form of maintaining higher diversity of practices in order to sustain segregation. See the
model of action choice along more than two dimensions in Online Appendix B. Bisin and Verdier (2000) also point out that effort
does not necessarily relate monotonically to outcomes: a small group could put in lots of effort to pass on their culture but it may
still die out, depending on the cost function.9Berman (2000) and Iannaccone (1992) model religion as a club good with ‘extreme’ cultural practices as a means of taxing
other goods to increase contribution to the religious good. This is slightly different to the framework presented in our paper and
the other papers cited, which draw on the Akerlof and Kranton (2000) model of identity.10We are reassured that our framework permits, as equilibria, outcomes consistent with those that are studied in detail in these
papers.
4
to adopt the same language or culture.11 In Lazear (1999) and Carvalho (2013a), agents choose
a cultural practice, but take interaction as given. A second literature models link formation
in order to study the important concept of homophily - the tendency for similar individuals
to be linked. In this case individuals choose interaction but take behavior as given (Currarini
et al., 2009; Bramoulle et al., 2012; Currarini and Vega-Redondo, 2011). We take a novel
approach which, importantly, addresses both these choices (choice of practices and choice of
interaction) within a single tractable framework.12 Our alternative approach allows payoffs
from links formed to depend not only on ex ante heterogeneity but also on the action choices
of both partners.13 This results in a broader range of social structures and comparative statics.
Our theoretical approach contributes to a literature on network formation where individuals
play coordination games across links, in particular Jackson and Watts (2002) and Goyal and
Vega-Redondo (2005). They show that in homogenous populations, under equilibrium refine-
ment, total integration and social conformism always prevail. We show that this is not true
for heterogenous populations in which different individuals have preferences for coordinating
on different activities. Our analysis of network formation in heterogeneous populations enables
us to study the important question of why different social structures prevail and when.
Card et al. (2008) and Chay and Munshi (2013), like us, test empirically for the presence of
a threshold in community behavior that depends on the size of the minority population, where
the location of this threshold is a priori unknown. Card et al. (2008) look at local migration
and Chay and Munshi (2013) at local migration and voting behavior. In contrast, our interest
is in social interaction and convergence (or not) of behaviors within the community. Together
with Card et al. (2008) and Chay and Munshi (2013), our findings highlight the importance
of looking for these kind of threshold patterns when considering community-related behavior.
Our findings are also in line with recent work on the US by Abramitzky et al. (2014), Fouka
(2014), and Fulford et al. (2015), which suggests that environmental features were significant
in determining how culture evolved in 19th and early 20th century United States.
In Section 2 we present the basic model. Section 3 characterizes the Nash equilibria for a
single generation and provides comparative statics. Section 4 develops the multiple generation
framework. Section 5 provides empirical evidence from communities in the United States in
the age of mass migration supporting the predictions of the model. The final section discusses
implications for government policy and welfare and concludes.
2 The Single Generation Model
First we present the framework for choice of action, then introduce social interaction, and finally
we summarize the payoffs.
11See also Eguia (2013) who examines this from the perspective of discrimination.12In a different framework, Bisin and Verdier (2000) highlight the importance of choice of social interaction and make segregation
effort a choice. However, they do not analyse resulting levels of segregation since their focus is whether diverse cultural traits
persist. Also Carvalho (2013b) examines a choice of segregation in the decision to veil or not.13Or, to put it the other way around, payoffs from actions taken to depend not only on ex ante heterogeneity but also on who
one linked with and their choice of action.
5
2.1 Culture
A population (or community) consists of a set of individuals i ∈ {1, 2, . . . , n}. Each individual
i is endowed with a type, k ∈ {M,m}, which is common knowledge. We refer to the majority
M types as ‘natives’, and the minority m as ‘immigrants’ (although many other interpretations
are clearly possible). There are nM ∈ N individuals of type M and nm ∈ N individuals of type
m, where nM , nm ≥ 2, n = nM + nm, and we assume nM ≥ nm.14
To illustrate the model with an example, consider the population to be a neighborhood. An
immigrant group has moved into the neighborhood from a different country and has come with
different cultural practices to the native group. We denote the cultural practices associated with
the native group by the action xM and the cultural practices associated with immigrant group
by the action xm.15 Cultural practices could be activities, such as type of food eaten or language
spoken, or behaviors, for example concerning education, gender, or marriage. Since cultural
practices are, in themselves, simply activities and behaviors, this leads to an observation: when
immigrants move to a new country they can, if they wish, adopt the cultural practices of the
native group (or vice versa). We refer to this as switching culture. While it is possible to switch
culture, there is a cost c from doing so. This is how we define culture: an individual chooses
an action xi from the set {xM , xm}, where it is possible to adopt the action associated with the
other group, but at a cost.16
Berry (1997) describes cultural changes that occur when groups with different practices share
the same environment as ranging ‘from relatively superficial changes in what is eaten and worn,
to deeper ones involving language shifts, religious conversions, and fundamental alterations to
value systems’. Regarding religion, Iannaccone (1992) highlights that ‘people can and often do
change religions or levels of participation over time’. The cost to switching culture can arise
for a variety of reasons. There may be fixed costs involved, such as learning a new language,
or participating in unfamiliar activities. Alternatively cultural practices may be considered
valuable in their own right.17 Culture can be so deeply entrenched that individuals find it
psychologically costly to adhere to behaviors or attitudes that differ from the culture one has
grown up with.18 We reduce these different possibilities to a single cost, c, in line with previous
work on culture choice (Akerlof and Kranton, 2000; Bisin and Verdier, 2000). The magnitude
of c is interpreted as a measure of cultural distance.
It is rather extreme to assume that all activities going on in this neighborhood are necessarily
associated with type or culture. For example, while religion often prescribes or prohibits certain
activities, religions rarely impact all aspects of everyday life. Religions rarely proscribe what
14We use the terms ‘native group’ and ‘immigrant group’ as an illustration. Of course we need not always consider the native
group as the majority group, for example Aboriginal populations of Australia and Native American populations of the United
States.15We model the practices that define a group’s culture by a single action. We enrich this model of culture by allowing for multiple
dimensions of culture in Online Appendix B. The main results remain.16The modeling of culture and a cultural group presented here is consistent with the introduction of identity into economic
modeling by Akerlof and Kranton (2000). Note that in our framework an individual chooses one action or the other. For some
practices such as language, however, an individual might continue to speak his group’s language, but also learn the language of the
other group. The results will hold provided that the relative benefit of learning the native language increases the more members of
the immigrant group there are that learn it. This would be the case, for example, if interaction and exchange among individuals
and groups occurs more and more often in the native language as more immigrant group members learn it.17Algan et al. (2013) estimate that the utility an Arabic parent in France gets from passing an Arabic name to their child is
equivalent to a 3% rise in lifetime income of the child. See also Bisin and Verdier (2000) and Kuran and Sandholm (2008) for
examples.18See Berry (1997) for a review and further references on psychological and sociocultural costs. This framework does not
incorporate group penalties, but such an assumption could be incorporated.
6
sports should be played. In a neighborhood with different religious groups it may be no more
costly (ex ante) for a member of either religious group to pick one sport as compared to another.
Which actions are specific to culture and which are not may vary by each particular situation.
Non-cultural actions are modeled by an individual i’s choice of action yi ∈ {yA, yB} for which
there is no associated type-specific cost.19 A word of caution: a result that we will highlight
later in the paper is that activities and practices that ex ante have nothing to do with type and
culture (for example, sports), can in equilibrium become associated with a particular group.
Thus many of the examples that spring to mind that associate sport or other activities with
a particular group may in fact be an equilibrium outcome and not a result of some ex ante
preferences.
To summarize the modeling of culture, each individual has a given type, M or m. Each
individual i of type k ∈ {M,m} chooses an action set (xi, yi), where xi ∈ {xM , xm}, yi ∈{yA, yB}. There is no type-specific cost associated with the non-cultural action yi whilst the
cost of cultural action xi is
ck(xi) =
{0 if xi = xk
c if xi 6= xk
2.2 Social Interaction
Were this the end of the model, individuals would never choose to pay the cost and switch
culture. However, there is another issue at stake. Social interaction within the population
is valuable, providing opportunities for economic exchange of varying types. Social ties (or
‘links’) allow for exchange of valuable information. For example, personal contacts play a large
role in information about job opportunities and referrals, suggesting an important effect of
social interaction on employment outcomes and wages (see Jackson, 2009, for a review of the
literature). Social interaction can also provide other economic opportunities, including trade,
favor exchange, or economic support such as risk sharing and other valuable joint endeavors
(Angelucci et al., 2012). We assume all individuals provide the same opportunities for economic
exchange.20
Crucially, personal connections and social interaction require commonality of some degree.
If two individuals do not speak the same language it limits exchange of information, discussion
and agreement on trade, and any other activities that involve verbal communication (Lazear,
1999). Diversity more generally has been found to reduce communication and interaction within
organizations (see Williams and O’Reilly, 1998, for a review). Communication difficulties aside,
if two individuals take part in completely different activities then not only do they rarely meet
(thus reducing opportunity for exchange), but even when they do meet they may not have
relevant information to exchange.21 There are two sides to this story: on the one hand a lack
of commonality can make forming a tie more costly or difficult, and, on the other, it will make
a tie less valuable if there are complementarities in information exchange and activities.
19Of course there may be multiple activities and behaviors associated with religion and multiple non-cultural activities. We
extend the cultural and non-cultural activities that can be chosen by the population to multiple dimensions in Online Appendix B.20There are two ways to relax this assumption in the current framework. The first is for one group to have greater opportunities,
so that, all else equal, members of that group would make a more valuable partner. The other is to include some degree of ‘love of
diversity’ to allow for benefits of getting different information and opportunities from different types. To introduce this one might
assume that the first tie with someone of a different type is highly valuable, while the value may decline the more contacts of that
type. The key trade-off of the model remains in place provided coordination remains important to interaction.21For further discussion on the need for coordination in interaction see Lazear (1999) and Kuran and Sandholm (2008).
7
We model this formally as follows. Individual i chooses whether or not to form a social
tie with the other n − 1 individuals in the population. If individual i forms a social tie with
individual j, we denote this by gij = 1, if not gij = 0. Player i’s choice of social ties can be
represented by a vector of 0’s and 1’s, gi = (gi1, gi2, . . . , gin), where gii = 0. If i forms a social
tie with j, the value of that social tie is increasing the more activities the two individuals have
in common. The value i receives from a social tie with j is
απ1(xi, xj) + (1− α)π2(yi, yj)− L,
where
π1(xi, xj) =
{1 if xi = xj0 if xi 6= xj
π2(yi, yj) =
{1 if yi = yj0 if yi 6= yj
and 0 < L < 1.
There is a fixed cost to forming a social tie, L. The benefit from a tie is increasing in
commonality of actions. Note that the parameter α measures the relevance of cultural versus
non-cultural actions in economic exchange. The framework can be interpreted in one of two
ways. Either, there is a fixed value normalized to 1 of having a personal connection with
another individual in the population, where the cost or difficulty of forming that tie depends
on how much the two individuals have in common. Alternatively, we can think of a fixed cost
L to forming a personal connection, where the possibility (or value) of economic exchange is
increasing the more individuals have in common.
2.3 Payoffs
Each individual i ∈ {1, .., n} chooses a cultural action xi ∈ {xM , xm}, a non-cultural action
yi ∈ {yA, yB}, and social ties gi = (gi1, gi2, . . . , gin) ∈ {0, 1}n. An individual’s strategy is thus
represented by the vector
si = (xi, yi, gi) ∈ Si. (1)
The utility of individual i of type k ∈ {M,m} is given by
uk(si, s−i) =∑j
(απ1(xi, xj) + (1− α)π2(xi, xj)− L)gij − ck(xi). (2)
The model presented describes an n-player game where each player i ∈ {1, . . . , n} chooses a
strategy si ∈ Si and receives a payoff uk(si, s−i). The strategy profile (s∗1, s∗2, . . . , s
∗n) is a Nash
equilibrium if uk(s∗i , s∗−i) ≥ uk(si, s
∗−i) for all i ∈ N and for all si ∈ Si. We refer to a strategy
profile (s1, . . . , sn) as a ‘state’. We examine pure strategy equilibria.
The set-up is akin to a version of a battle-of-the-sexes game between groups. An individual
gets a higher payoff the better he coordinates with his social ties, allowing for greater economic
exchange. But, he would rather coordinate on his own cultural practices. The framework
has the additional twist that each individual chooses his social ties. An individual also gets
a higher payoff the more social ties he has with whom he is coordinating, again allowing for
greater economic exchange. The assumption here, that more contacts are better, has support
from the literature on social interaction (Currarini et al., 2009; Currarini and Vega-Redondo,
8
2011). It implies that individuals might trade-off cultural costs against the benefits of more
contacts.22
Note two things. Individual i cannot differentiate his choice of action by social tie. Intu-
itively, this requires that an individual be ‘consistent’ in his behavior within the given popula-
tion.23 For example, it is often not possible to adopt two different religions. Religious activities
might take place at the same time, the different practices might be contradictory, it could be
too time-consuming to do both, or it might be made impossible because of hostility from others.
Second, social ties are one-sided. If i forms a social tie with j then the value of the social tie
accrues to individual i but not to individual j unless individual j forms a social tie with i. In
equilibrium, however, if i forms a social tie with j then j will also form a social tie with i. The
model of link formation presented here is not the only way to model social interactions and is
chosen to simplify the exposition (see Online Appendix B for further discussion).24
3 Analysis of the Single Generation Model
In this section we characterize the Nash equilibria of the game and provide comparative statics.
We first introduce two assumptions made to simplify the exposition. Assumption 1 rules out
the uninteresting case where a single individual prefers to maintain his cultural action even if
the rest of the population all adopt a common, different cultural action.
Assumption 1 (No Man is an Island)
(1− L)(n− 1)− c > max{0, (1− α− L)(n− 1)}.
Second, we assume that when an individual is indifferent between forming a tie or not, the tie
is formed.25
3.1 Characterization of Nash Equilibria
Proposition 1 describes the Nash equilibria of the game. Nash equilibria take one of three
contrasting forms: (i) assimilation, (ii) segregation, or (iii) multiculturalism. Nash equilibria
are characterized by thresholds on the share of immigrants in the population: if the immi-
grant group is small, assimilation states are the only equilibria; if the immigrant group forms a
large enough share of a population, different social structures can emerge in equilibrium. For
convenience we first introduce our formal definition of assimilation, segregation, and multicul-
turalism.
Definition 1 A state s = (s1, . . . , sn) is defined as:
Assimilation if all individuals adopt the same actions, xi = xj, yi = yj ∀i, j ∈ N , and
each individual forms a social tie with all other individuals.22We discuss alternative specifications of this assumption in Online Appendix B.23A population consists of n individuals who each have the opportunity to interact with all the others. Thus a population refers
to a workplace, a neighborhood, a school, etc. An individual may interact in multiple populations and play different strategies in
different populations. This framework then does not prohibit different behaviors at work or school from those in the neighborhood,
for example.24The Nash equilibria in this paper all satisfy the definition of pairwise stability, an important measure of stability in network
formation (Jackson and Wolinsky, 1996), although these are not the only pairwise stable outcomes. Alternatively, a two-sided link
formation model, related to pairwise stability, where an individual could delete any number of links and form any number of agreed
upon links would produce the same outcomes as the link formation model we use.25Both can be relaxed.
9
Segregation if type M adopt action xM , type m adopt action xm, and each individual
forms a social tie to all other individuals of the same type as him but does not form a
social tie to individuals of a different type.
Multiculturalism if type M adopt action xM , type m adopt action xm, both types adopt
the same non-cultural action yi = yj ∀i, j ∈ N , and each individual forms a social tie to
all other individuals.
Proposition 1 States which satisfy the definition of assimilation, segregation, and multicul-
turalism are the only possible Nash equilibria of the game. Further,
(i) Any assimilation state is always a Nash equilibrium;
(ii) There exists a Nash equilibrium which satisfies the definition of segregation if and only if
the share of the minority group in the population weakly exceeds δ;
(iii) There exists a Nash equilibrium which satisfies the definition of multiculturalism if and
only if the share of the minority group in the population weakly exceeds η;
where
δ = max
{1
2− c+ L− 1
2n(1− L),
1− α− L2(1− α)− L
+1− α
(2(1− α)− L)n
}, η =
{12− c−α
2nαif 1− α ≥ L
1 if 1− α < L
The proof is found in Appendix A. Only three types of social structure are Nash equilibria.
Assimilation: one group pays the cost of switching cultural action to facilitate interaction with
the other group. Segregation: groups do not pay the cost of switching cultural action but
restrict interaction within their own group. And a third structure, multiculturalism, in which
groups each maintain their respective cultural action but adopt a common non-cultural action.
Crucially, coordination on these non-cultural practices enables interaction across groups.26
In our framework, an individual’s payoff from taking an action is weakly increasing in the
number of others that take that same action. It is important to highlight the role that such
strategic complementarities play in our findings. First, this interaction between the individual’s
payoff and what the rest of the community are doing gives rise to multiple equilibria, allowing
similar populations to end up in contrasting states. To see this, suppose individual i’s group
assimilates, so everyone in the population adopts the other group’s cultural practices. Main-
taining his own cultural practices then becomes very costly to i since it hampers interaction
with everyone else, thus he can do no better than assimilate also. Hence assimilation is clearly
an equilibrium. Now consider the other extreme: suppose everyone in i’s group segregates and
maintains their own cultural practices. Maintaining his own cultural practices is now much less
costly to i, since he can still interact fruitfully with everyone in his group. Thus segregation
may also be sustainable as a Nash equilibrium. Second, strategic complementarities create a
26These three distinct forms of social structure are evident in scholarly discussion of communities in the United States following
mass migration (Gordon, 1964). Assimilation among immigrants in certain communities persuaded early scholars that a single
‘American culture’ would prevail. As it became clear that not all communities assimilated, but some ‘retained distinctive economic,
polical and cultural patterns long after arriving in the United States’, scholars accepted that assimilation was not the only possibility
for diverse communities (Bisin and Verdier, 2000). Even early on, however, there was discussion of a ‘third way’ where immigrants
could become ‘American’ and integrate but also maintain some cultural distinction.
10
nM
nm
(A)
(B)
(C)
Figure 1
nM
nm
n
δnηn
Figure 2
The Nash equilibria are illustrated for parameter values 1 − α ≥ L. The axes measure the size of each group. Group
M is only in the majority above the 45◦ line, so the area below the 45◦ line is greyed out. Assimilation is a Nash
equilibrium in areas (A), (B), and (C) in Figure 1. Multiculturalism is an equilibrium in areas (B) and (C). Segregation
is an equilibrium in area (C) only. Figure 2 illustrates the same graph highlighting the results for a population of
fixed size n. The dashed line shows all possible shares for the minority group, from nm/n = 0 to nm/n = 1/2, for a
population of a given size n. The dotted lines illustrate, for this population of size n, the size of the minority group
above which multiculturalism and respectively segregation are equilibria. Note, the size of a minority group that can
sustain multiculturalism is smaller than the size of the minority group that can sustain segregation because the cost to
the minority group of multiculturalism is less than the cost of segregation.
multiplier effect, driving members of a group towards doing the same thing.27 Even if we add
some within group heterogeneity – see Online Appendix B for this addition to the model –
because of the multiplier effect members of the same group will still have a tendency to do the
same thing.
The two features discussed above lead to the key finding that equilibria are characterized by
thresholds on the share of the minority group. When the minority group is small, assimilation
states are the only equilibria; segregation and multiculturalism are equilibria only in populations
where the share of the minority group is above the thresholds given in Proposition 1. An easy
way to see this is to observe that maintaining a distinct culture is only ‘worth it’ if enough
others do so as well. Segregation is an equilibrium when no individual in the minority group is
willing to pay to switch culture and interact with the larger majority group, which is true only
when the share of the minority group is large enough. Thus only if the share of the minority
group reaches this ‘critical mass’ or ‘threshold’ can segregation be sustained. An analogous
intuition holds for multiculturalism.28 At these critical masses, a small change in the share
of the minority group can result in a large change in equilibrium social structure. This is
illustrated in Figures 1 and 2.
27By multiplier effect we mean the following. Once one individual adopts a particular action this raises the relative payoff to that
action, which may then induce other agents to adopt the action, which further raises the payoff to that action, which may induce
further agents to adopt, and so on.28Under multiculturalism the minority group will interact with the majority group, but the benefit of interaction is lower than if
they were to assimilate, and therefore the same intuition holds.
11
3.2 Comparative Statics
The parameters c, α, and L shift the location of the thresholds described in Proposition 1. That
is, c, α, and L determine the size of the critical mass of immigrants that is necessary to sustain
segregation or multiculturalism. The parameter L is the cost of forming a link. Parameters c
and α both measure ‘culture’ but have distinct interpretations. The magnitude of c is the cost of
adopting the other group’s cultural action. For example, it might be less costly to switch from
speaking German to English, which is a closely related language, than it would be to switch
from Italian. Contrast this with the parameter α, which measures the importance of cultural
activities relative to non-cultural activities in everyday interaction and economic exchange. A
high α environment is one in which cultural practices are frequently relevant to interaction. For
example, a environment in which most social activities taking place are organised by churches
and other religious organizations. In this example, a low α environment is one in which social
activities related to religion comprise only a small part of daily life.
Corollary 1 The threshold share of the minority group above which segregation states are Nash
equilibria, δ, is decreasing in cultural distance, c; decreasing in the importance of culture, α;
and decreasing in the cost of forming a link, L.
The threshold share of the minority group above which multiculturalism states are Nash equilib-
ria, η, is decreasing in cultural distance, c; but increasing in the importance of culture, α; and
increasing in the cost of forming a link L.
When cultural distance between groups, c, is higher, switching culture is more costly, and
so smaller minority groups are more willing to maintain their own culture. This makes it easier
to sustain both segregation and multiculturalism and the respective thresholds both fall.
Counterintuitively, an increase in the importance of culture, α, makes groups less willing to
maintain their own culture under multiculturalism (the threshold rises). This is because, under
multiculturalism, a higher α makes having a common culture more important to interaction,
so maintaining one’s own culture is more costly in terms of lost opportunities for exchange.
Indeed, when culture ‘dominates’ everyday life (precisely α > 1 − L), social interaction based
on common non-cultural actions is not enough to sustain a tie and multiculturalism breaks
down. To see this, consider again an environment where most social activities are related
to religion. It is then difficult for individuals to maintain their own religious practices and
integrate with those who adopt different practices. The set of equilibria may be reduced to just
segregation and assimilation. An increase in α has the opposite effect on segregation making it
a less costly option. This is illustrated in Figures 3 and 4.
A higher cost of forming a link, L, makes it easier to sustain segregation (lowers the thresh-
old), by making assimilation a less attractive option for minority members. The costs of assim-
ilation come from switching culture, while the benefits come from improved interaction with
natives; a higher cost of link formation lowers the relative benefits of interaction with natives.
In contrast, a higher cost of forming a link makes it harder to sustain multiculturalism. When
the costs of forming a link are high, these can outweigh the benefits of interaction with the
other group under multiculturalism.
12
nM
nm
(A)
(B)
(C)
Figure 3: An increase in c.
nM
nm
(A)
(B)
(C)
Figure 4: A reduction in α.
Figure 3 replicates Figure 1 and illustrates an increase in c. The arrows show that areas (B) and (C) expand with an
increase in c. That is, the parameter ranges under which segregation and multiculturalism are equilibria expand with an
increase in c. Figure 4 replicates Figure 1 and illustrates a reduction in α. The arrows show that area (B) expands and
area (C) shrinks with a reduction in α. That is, the parameter range under which multiculturalism is an equilibrium
expands with a reduction in α and the parameter range under which segregation is an equilibrium shrinks.
To summarize, we find that one of only three structures – assimilation, segregation or mul-
ticulturalism – can arise in heterogeneous communities in equilibrium. Equilibria are charac-
terized by a threshold, or ‘critical mass’. When the minority group is small, assimilation is the
only equilibrium. The size of the critical mass of immigrants necessary to sustain segregation
or multiculturalism depends on cultural distance, the importance of culture in everyday life,
and the cost of link formation. In Section 5 we find evidence of such thresholds in heteroge-
neous communities in the United States in the age of mass migration: if immigrant groups hit
a critical mass in the local community (the magnitude of which we estimate) they are much
more likely to maintain their own practices and segregate.
A low importance of culture, α, appears at first to be a positive force for cross-group inter-
action since it makes it easier to sustain multiculturalism. However, our second comparative
static result shows that a lower importance of culture leads to more polarizing behavior in
segregation equilibria.
Corollary 2 When α ≤ 1− L segregation is a Nash equilibrium when nm/n ≥ δ if and only if
groups adopt different cultural actions and different non-cultural actions.
When the importance of culture is low, α ≤ 1 − L, the segregation Nash equilibria require
that the two groups adopt different cultural actions and different non-cultural actions. Groups
differentiate their practices above and beyond ex ante cultural differences. This is not simply a
result of coordination on different non-cultural actions. Instead the minority group must adopt
a different non-cultural action y from the majority, in order to raise the cost of interaction with
the majority group such that segregation can be sustained. This outcome is an equilibrium,
despite the reduction in economic exchange that it entails. We refer to this type of segregation
equilibrium as ‘extreme segregation’.
13
Emergent Cultures
Non-cultural actions, that is activities and practices with no type-specific costs, play an impor-
tant role in the social structures that emerge in equilibrium. It has long been recognised that
when groups with distinct cultures interact, individuals do not simply choose between which
of these cultures to practice. Instead, interaction can produce new ‘emergent cultures’, where
actions that were previously culturally inconsequential take on an important role in uniting
or dividing the community. This result can be seen clearly in the equilibria produced in our
model.
In the multiculturalism equilibrium, groups maintain their different cultural actions, but the
two groups adopt a common non-cultural action, either yA or yB. Construction of common
practices is necessary to sustain integration when groups also retain some diversity of practices.
This equilibrium can be related to the frequently advanced ideal that populations can maintain,
for example, their diverse religions while at the same time having a common national identity
and culture.29 We show that this ‘ideal’ is feasible when culture is not ‘all encompassing’ in
society; groups must be able to find enough common ground to be able to interact.
At the other end of the spectrum, under the extreme segregation equilibria, practices can
become more polarized: groups differentiate their behaviors and practices beyond different
cultural actions. Harris (2009) writes that some minority groups hold ‘secondary cultural
differences . . . that emerge after the two groups have been in continuous contact.’ We show that
when α is low, segregation can be maintained only when each group coordinates on distinct non-
cultural actions. Rather than bridging the gap between groups, the non-cultural action is used
to emphasise differences between them. For example, groups practising two different religions
ex ante will also differentiate other behaviors that are un-related to religious requirements (for
example, dress code, increased food restrictions, and even sports).30 This appears to be a novel
explanation of this type of behavior and we find it to be a positive signal of a model purporting
to explain cross-cultural interaction that there is an equilibrium admitting this possibility.31
4 Multiple Generations with Cultural Transmission
Section 3 characterizes the equilibria for a single generation. In this section we incorporate the
framework into a model of cultural transmission over generations. Considering multiple genera-
tions is important for two reasons. First, it may be that some outcomes that are Nash equilibria
in a single generation are likely to die out in the long run. Second, since the static model has
multiple equilibria for a given configuration of parameters, it is useful to consider whether some
Nash equilibria are more ‘likely’ than others, thus the multiple generation framework acts as
29As early as 1915, at a time of fervent discussion of integration of immigrants in the United States, Harvard philosopher Horance
Kallen described the possibility of the United States being ‘a democracy of nationalities, cooperating voluntarily and autonomously
through common institutions’ where ‘the common language . . . would be English, but each nationality would have . . . its own peculiar
dialect or speech, its own individual and inevitable esthetic and intellectual forms.’30Berman (2000) describes the birth of Ultra-Orthodox Judaism in the late 18th and 19th Centuries which followed emancipation
and the possibility of greater integration with the local European populations. The Ultra-Orthodox ‘were not only conservative
about rejecting new forms of consumption . . . but amplified existing restrictions’, such as introducing new dietary restrictions, and
‘changed existing customs (dress codes, speaking Yiddish) into religious acts’.31A number of papers examine this type of outcome in detail including Berman (2000) on Ultra-Orthodox Judaism; Austen-
Smith and Fryer (2005) on why some black students deride working hard at school as ‘acting white’ and act in opposition to this;
and Akerlof and Kranton (2002) and Bisin et al. (2011) on the emergence of oppositional identities, whereby groups increase their
identification with their own culture in order to reduce the psychological cost of interacting with those who adopt a different culture.
14
an equilibrium refinement.
We incorporate dynamic features into the single-generation model. First, we assume a
multiple generation framework where parents pass on their type to their children (a simplified
version of Bisin and Verdier, 2000). Second, we do not assume that the population automatically
starts off at an equilibrium. Children, once born, choose their actions and social ties optimally
as above, but taking as given the existing choices of the population they are born into. This
means that all else equal, a child born into a more assimilated group may behave differently
to one born into a more segregated group, again capturing the importance of the behavior
of the rest of the community. Finally, we allow for a few members of a new generation to
‘experiment’ or ‘make mistakes’. We can also think of these as temporary shocks which happen
infrequently. For example, suppose the population is in a state where members of the minority
group maintain their minority culture. In the next generation, however, a few children in the
minority group ‘experiment’ and try out majority culture. This can set in motion a move to a
different Nash equilibrium: if enough people experiment it may be optimal for future minority
group children to also adopt majority culture. These shocks allow populations to move away
from ‘less stable’ Nash equilibria and towards ‘more stable’ ones.32
The dynamic is modeled formally as follows. Suppose at (discrete) time t the population of
size n is in some state which is described by (st1, st2, . . . , s
tn), where sti is the choice of actions
and social ties of individual i at time t. A strategy at time t, sti, is given by (1), as in the
single generation case. The state need not be a Nash equilibrium. At time t + 1 there is an
independent probability q ∈ (0, 1) for each individual i (parent) that they die and are replaced
by a child, also denoted i. The child is the same type as the parent, M or m, and chooses an
action which maximizes his utility given the state in period t:
st+1i ∈ arg max
Si
uk(st+1i , st−i)
where uk(st+1i , st−i) is given by (2). Under this dynamic the population moves to a Nash equilib-
rium, and once it reaches an equilibrium it will stay there for all future generations. However,
we also allow that with probability ε the child instead adopts some other strategy randomly.
This is how we model experimentation by children.
We examine the case where the probability of experimenting, ε, is small. The process
detailed above defines an aperiodic and irreducible Markov chain. The following proposition
tells us that, for any given parameter values, a single outcome will be observed almost all
of the time.33 Which outcome is observed in the long-run depends on the parameters of the
population in a similar way to the single-generation case. We make the following assumption,
which ensures cultural distance is sufficiently important that the trade-off between maintaining
one’s own cultural practices and forming social ties is economically relevant.34
Assumption 2 c ≥ (1− L)(nm + 1)
Under Assumptions 1 and 2 we get the following result.
32As described in Young (1993), the process we model ‘selects the equilibrium that is easiest to flow into from all other states
combined, including both equilibrium and non-equilibrium states’.33There is a unique stationary distribution which puts all weight on a single state.34It is made for ease of exposition. If Assumption 2 does not hold, then the cost of switching culture is low, and both assimilation
to M or assimilation to m could persist in the long run. See the proof of Proposition 2 for details.
15
Proposition 2 When the probability of experimenting is small, ε→ 0, and t→∞, the follow-
ing outcome is observed with probability tending to 1:
(i) When 1− α < L there exists a threshold δ∗ such that
Assimilation by the minority group is observed if nm/n ≤ δ∗ − 1/n;
Segregation is observed if nm/n ≥ δ∗ + 1/n.
(ii) When 1− α ≥ L+ ( 1nM−1
) there exists a threshold η∗ such that
Assimilation by the minority group is observed if nm/n ≤ η∗ − 1/n;
Multiculturalism is observed if nm/n ≥ η∗ + 1/n.
In the long-run, a unique outcome emerges (or is more likely to emerge). Proposition 2 tells
us that if the minority group is small, assimilation occurs and it is the minority group that
assimilates. It also shows that multiculturalism can be sustained as a stable long-run outcome
in a diverse population. This occurs when culture is not too important in everyday life (α low)
and the minority group is large. If culture is very important (α high) and the minority group
is large, then we will see long-run segregation of groups.
Analogous to the single-generation model, higher cultural costs, c, allow populations with
a smaller proportion of minority individuals to sustain segregation or multiculturalism in the
long run. A lower importance of culture, α, can ensure multiculturalism is a long-run outcome
instead of segregation, and allows multiculturalism to be sustained in populations with a smaller
proportion of minority individuals. The result that different cultural practices can persist, under
certain parameters, even in the long-run, is consistent with the findings and discussion in Bisin
and Verdier (2000).
The proof of Proposition 2 involves assessing the minimum number of shocks needed to
induce a transition from one Nash equilibrium to another, and then aggregating to determine
which of the equilibria is most likely to occur in the long-run. We simplify this by using a ‘tree
pruning’ argument, the details of which can be found in the proof in Appendix A. The +1/n
and −1/n terms added to the thresholds and the small positive term +( 1nM−1
) in Proposition
2 are there to avoid having to make a more complex statement about what happens at the
boundary parameters between one long-run outcome and another. Details and precise values
for thresholds can be found in Appendix A and Online Appendix B.
The forces driving the results of Proposition 2 are intuitive. First, why does the minority
group assimilate and not the majority group? Suppose in generation t the minority group and
majority group adopt their respective cultural practices. In generation t+ 1 some members of
the minority group experiment by adopting majority culture and interacting with the majority
group. If enough members experiment, then future minority group members will choose to
switch and also adopt majority culture. Since the minority group is smaller, less experimenta-
tion is required to induce further minority group members to optimally choose to assimilate to
majority culture than if we repeat the same scenario with the majority group. For a similar
reason, assimilation occurs only when minority individuals constitute a small share of the pop-
ulation: the smaller the minority group, the less experimentation it takes to induce movement
in the minority population towards assimilation.
16
A natural extension of the multiple generation framework would allow the costs of practices
to change over generations. If a parent assimilates, the cost to the child of switching culture may
be less than the cost the parent faced. Alternatively, where a parental generation segregates
and further differentiates non-cultural actions, the child may face type-specific costs over the
actions for which the parents faced no cost.35 This would be an interesting avenue for future
research.36
5 Evidence from the Age of Mass Migration
The ‘Age of Mass Migration’ in the late 19th and early 20th centuries is one of the most
important episodes of migration of the modern era. From 1830 to 1930, 38 million people
arrived in the United States. They joined a population that in 1830 consisted of only 13
million people, and which Alba (1985) describes as having both ‘culture and institutions, [that]
derived largely from the English models’. Immigrants came to America from around the world,
including virtually every country in Europe.37
We look at heterogeneous communities throughout the United States in this era and ask how
this heterogeneity manifested itself. Did immigrants adopt the behaviors and practices of the
local population? Did they integrate or form segregated communities? The setting provides a
natural environment in which to examine some of the implications of our model. Importantly, it
provides us with a large number of heterogeneous communities with different immigrant groups
of varying sizes.
We test two key predictions of our model: that group behavior changes sharply once the share
of the immigrant group in the community reaches some (ex ante unknown) critical threshold,
and that the location of this threshold varies in the predicted way with the cultural distance
between immigrants and natives. In particular, we consider two choices facing immigrants:
speaking English, and in-marriage (homogamy).38
Testing these predictions of our model is important for two reasons. First, if our framework
captures an important trade-off, then we should expect to observe such thresholds in data.
Second, evidence of such thresholds, and hence of the forces which drive our model, has many
implications for policy.
5.1 Data Description
Data Sources
We use data from the 1900, 1910, 1920, and 1930 census samples provided by IPUMS. These
are 5% (1900 and 1930) or 1% (1910 and 1920) samples. These censuses asked individuals
35Observe, it is not the case that cultural costs necessarily fall over generations. Suppose one individual from the parent generation
experiments and assimilates while the rest of his group segregates. Then the cost to his child of speaking the other group’s language
falls somewhat. However that child may still find it preferable to choose to segregate, in which case the costs of speaking the other
group’s language may go up again for the grandchild.36One way to do this would be to combine the current framework with features of Kuran and Sandholm (2008), which allows
costs of taking an action to change as actions taken change.37Not all immigrants remained in the US permanently (Bandiera et al., 2013). We will discuss later the implications of this for
our results.38In contrast to our baseline modeling assumption, immigrants could continue to speak both their native language and learn
English. As mentioned previously, our results will hold provided that the relative benefit of learning English increases the greater
the number of members of the immigrant group who learn it.
17
whether or not they speak English. They also allow us to link (almost all) married individuals
to their spouses, so we can measure whether individuals marry endogamously (i.e. within the
same nationality).39 Important to our question, in these particular years individuals were asked
their country of origin, and – unlike earlier and later censuses – their year of arrival in the US,
allowing us to control for how long individuals have lived in the US. Focusing on adult household
heads, we have a repeated cross-section with 611,000 migrants.
The census samples provide information on an individual’s place of residence down to the
county level. Counties are small administrative units, with on average 63 counties per state,
and an average population of 5300 households per county in 1900, of which 1300 were headed
by immigrants.40 We also use data from Spolaore and Wacziarg (2009), collected by Dyen et al.
(1992) and Fearon (2003), on the linguistic distance between pairs of languages.
Data Construction
We treat counties as the relevant population for our analysis. That is, individuals are presumed
to make their decisions about what behaviors to adopt and with whom to form ties, taking those
in the county as the pool of people they could potentially interact with.41
Within each county, we define an immigrant group as consisting of all household heads of
the same nationality (country of origin), with the exception that we group Germany, Austria,
and Switzerland into a single group; Sweden, Denmark, and Norway into a single group; and
the Netherlands and Belgium into a single group.42, 43 Our qualitative results are unchanged
in the absence of this aggregation. Henceforth, by ‘nationality’ we refer to these groups. In
each county, the ‘nationality share’ is the proportion of sampled adult household heads in that
county who have that nationality: this is the empirical analogue of nm/n in our model.44
Since our model predictions are at the group level, we aggregate the behavior of individuals
into ‘cohorts’. Cohorts are defined by nationality, d; year of arrival (in 10 year bands), a; time
since immigration (also grouped, to the nearest 10 years), t; and county, o.45 Splitting the
39Individuals whose spouses do not live in the same household cannot be matched. They make up less than 3.1% of married
individuals in our sample.40Note, the difference between these figures and Table 1 comes both from changes over time in county size (the table pools all
censuses), and from the fact that larger counties have more observed cohorts, so receive more weight in the table.41Counties are the finest population partition available to us. It seems possible that decisions are made based on a more local
population of people, such as the town/village. In that case we observe only a noisy measure of the population share and population
decisions, likely attenuating our estimated effect and reducing power.42We focus on household heads since they were less likely to have received education in the US which would change their decision-
making. At the time they also were likely to have made choices for the whole household. In the case of in-marriage it also avoids
the double counting of homogamous relationships.43We made these groupings due to use of a common language and because accounts of communities in the US suggest these
amalgamations are appropriate. Austrians spoke, in main part, a form of German, and the Swiss who emigrated were largely
German speaking. At the time we are considering, Danish remained the official language of Norway, and also Swedish, Danish,
and Norwegian are considered mutually intelligible. The Belgians who emigrated to the US were largely of Flemish descent, and so
spoke Dutch.44Note that this treats all individuals who are not members of that nationality group as though they were members of the
majority group. In all counties, the largest other group that immigrants could consider joining is the native group. Hence, if an
immigrant group is considering switching to another culture, this is likely to be the most profitable. Although in some cases it may
be lower cost to join another immigrant group due to their lower cultural distance, this is likely mitigated both by our amalgamation
of individuals from certain countries and the fact that these groups are typically much smaller than the natives. In our analysis
sample there are only 33 counties where more than one group exceeds 10% of the county population, 4 counties where more than
one group exceeds 15%, and none at a threshold of 20%. An alternative empirical specification would be to define the denominator
for nationality share as the sum of the number of immigrants of that nationality and natives, thus excluding other immigrants. Our
results are robust to this redefinition.45In particular, arrival years are grouped as 1886-1895, 1896-1905, and so on. Tenure (time since immigration) is then measured
as the difference between the midpoint of the arrival year band (1890, 1900, etc) and the census year.
18
observed behavior of immigrant groups into these cohorts allows us to control for arrival year
and tenure effects that might be important in explaining behavior and might vary systematically
by nationality. For example, both ability to speak English and out-marriage are likely to be
positively related to tenure in the United states. Additionally, varying conditions and policies
over the decades we study, such as use of schooling to influence assimilation (Bandiera et al.,
2015), may have affected these decisions.
We measure English acquisition by the proportion of a cohort that reported to the enu-
merator that they spoke English.46 For this analysis we exclude British, Irish, and Canadian
immigrants, since their mother tongue is likely to be English anyway. To proxy for social seg-
regation, we construct a measure of the extent to which each cohort marries people of the same
nationality. We calculate the proportion of married members of the cohort that are married to
someone of the same nationality.47 Although marriage is a partial description of social connec-
tions, it is clearly defined and measured for this period, and marriage decisions are considered
to be strongly revealing of the dimensions that divide a society (for a review see McPherson
et al., 2001).
Table 1 shows the mean and variance for these outcomes, as well as nationality size (nmin our model), county size (n), and nationality share (nm/n), across the analysis sample. We
also provide the within-nationality variance, and the share of total variance that is within
variance. From this it can be seen that there is significant within nationality variation that we
can exploit in our analysis, allowing us to avoid concerns that the effects we capture come only
from differences across nationalities.
To measure the cost of learning English, c in our model, we use the data on linguistic distance.
The data from Dyen et al. (1992) define linguistic similarity between a pair of languages based
on the proportion of frequently used words that share a common root.48 We assume that
immigrants’ mother tongue is the majority language spoken in their country of origin. We also
assume that the cost of switching language to English is greater for those immigrants whose
mother tongue is more dissimilar to English. We then split cohorts into ‘near’ or ‘far’ from
English, defining a language as far from English if less that 40% of the words are mutually
intelligible with English.49
For our analysis we also only consider cohorts with at least 30 observations, so that the
average group behavior is well estimated. This means that we exclude counties where the
number of immigrants was very small. There is a trade-off between choosing a low minimum
threshold for immigrant group size, to minimise the bias that comes from effectively ignoring
very small community shares, and choosing a high threshold to reduce noise in estimated group
46Enumerators were instructed ‘Where difficulty is encountered in making the head of the family understand what is wanted, you
should call upon some other member of the family who is able to speak English . . . if no member of the family can aid you in your
work, then the assistance of some neighbor of the same nationality and able to speak English should be obtained, whenever possible.’
Where this was also not possible, paid interpreters were used. Hence this question is likely to have been able to meaningfully capture
whether individuals were able to speak English. https://www.census.gov/history/pdf/1900instructions.pdf47The Coleman index provides an alternative popular measure of segregation (see Currarini et al., 2009). It is not suitable for
our context as it requires knowledge of the potential marriage market for a particular cohort at the time marriage decisions were
taken, something we do not observe. Our preferred measure is therefore the simple ratio of in-marriage to total marriage for a
cohort, which has the additional benefit of being easy to interpret.48This measure of distance is only available for Indo-European languages. We impute some of the missing data using a variable
from Fearon (2003) which measures linguistic proximity using a ‘language family tree’. Our results are qualitatively robust to
instead dropping these observations with missing data.49We take this cut-off from Advani and Rasul (2015), who choose it because there are no languages in the range 30−50% mutually
intelligible with English, so it forms a natural break. According to this metric, Spanish and Greek are considered far from English,
whilst German and Danish are considered close.
19
Table 1: Descriptive statistics for key variables
(1) Mean (2) Total Variance (3) Within Var (4) Within/Total
Share speaking English .901 .023 .011 .489
Share of in-marriages .609 .055 .022 .391
Nationality Size (’000s) .459 .391 .374 .957
County Size (’000s) 8.79 106.9 96.5 .903
Nationality Share .085 .007 .005 .732
Observations 1746 1746 1746 1746
Notes. Observations are at the cohort level. Cohorts are defined by nationality, county, tenure in the US grouped
to the nearest 10 years, and year of arrival in 10 year bands. We include only cohorts with at least 30 individuals,
to ensure that cohort averages are well measured. Share speaking English measures the proportion of a cohort that
reports speaking English. Share of in-marriages measures the proportion of married individuals in a cohort who
are married to someone of the same nationality. Nationality size measures the number of people, in thousands, of
the same nationality (potentially of different cohorts) living in the same county. County size measures the number
of people, in thousands, living in the same county. Nationality share is the ratio of nationality size to county size.
Columns (1) and (2) provide the mean and variance for these variables. Column (3) shows the remaining variance
after removing nationality fixed effects, and Column (4) provides the share of total variance that remains once
nationality fixed effects are removed.
behavior. We test the sensitivity of our results to this by also using cut off points of 20 or of
40, and find no qualitative difference. As an alternative sensitivity test, we also performed our
analysis with weighted regressions. In particular, the weight given to an observation was equal
to the number of observations with the same nationality share in the full sample, divided by the
number with that nationality share in the estimation sample.50 This over-weights observations
with small nationality shares, since these are the ones excluded by our group size thresholds.
Again we find similar results. These results are presented in Online Appendix C.
In total we have 1746 cohorts, when we impose a minimum cohort size of 30, of which
Germans and Italians are the biggest groups. Table C1 in Online Appendix C shows the
number of immigrant cohorts we observe, and splits out some of the larger nationalities. It also
shows how this varies depending on the minimum size we require for a cohort to be included
in our sample.
5.2 Empirical Framework
To formally test for the existence of a threshold in community behavior we use a linear regression
of the form
Ydato = β0 + β11{zdo > τ}+ γ0zdo + γ1zdo1{zdo > τ}+ κd + λa + νt + εdato (3)
where Ydato is the share of immigrants of nationality d, arriving in year a, with tenure t, in
county o, who speak English (respectively, in-marry); zdo is the nationality share of immigrants
of nationality d in county o; and τ is the proposed threshold immigrant share. κ, λ, and
ν are respectively the nationality, arrival year, and tenure fixed effects. The coefficient of
interest is β1: the coefficient on the indicator function showing whether there is a ‘break’ in the
level of speaking English (respectively, in-marriage) when the nationality share zdo crosses the
threshold τ .50For constructing weights we split nationality share into deciles, with all observations in the decile being given equal weight.
20
Figure 5: Share of people in the cohort that speak En-
glish
Figure 6: Share of married people in the cohort that are
married to someone of the same nationality
Notes. The figures show the relationship between the share of a cohort that speak English (Fig 5) or marries within its nationality
(Fig 6), and the proportion of adult household heads in their county that are of the same nationality. Dots show the mean share
speaking English/in-married, grouping nationality share into bins two percentage points wide. The line shows the kernel-weighted
local mean, estimated using an Epanechnikov kernel with bandwidth of 7% fit through the whole (i.e. unbinned) data. In each
figure the local mean is estimated either side of an estimated threshold: .31 for Fig 5 and .35 for Fig 6. These points are chosen
using the Quandt Likelihood Ratio procedure descibed in Section 5.2. Cohorts are defined by nationality, 10 year grouped arrival
year, 10 year grouped tenure in the US, and county of residence. A minimum cohort size of 30 is used.
Since we do not know theoretically where the threshold τ should be, we use an iterative
regression procedure, and test for significance using a Quandt Likelihood Ratio (QLR) test
(Quandt, 1960; Andrews, 1993). We perform a sequence of regressions, testing a prespecified
range of values for τ . For each regression we calculate the F-statistic, comparing the model
with the threshold to the same model but without a threshold. We then select from among
these regressions, the one with the highest F-statistic. The corresponding break point in that
regression is then taken as the estimated location of the threshold, denoted τ ∗. To test whether
this threshold value is ‘significant’, we compare the F-statistic to the limiting distribution for
this statistic under the null (Andrews, 1993), thus correcting for the multiple testing.51, 52
We also estimate Equation 3 separately for cohorts that are ‘far from’ and ‘near to’ English,
and compare the point estimates, to see whether τ ∗far < τ ∗near, for both speaking English and
in-marriage. However, we are unaware of any formal way to test for the statistical significance
of any difference in location.
5.3 Results
Threshold in proportion speaking English
Figure 5 shows graphically the evidence for a threshold in the share of the immigrant group
that speaks English, as its share of the community rises. Each dot shows the mean proportion
51For robustness, we also use an entirely non-parametric approach to finding the threshold location, as proposed by Henderson
et al. (2015). The intuition of this approach is similar to the parametric approach, searching over a sequence of thresholds, but
without any functional form restrictions away from the threshold. The location of the thresholds we find are the same as for the
parametric approach.52An alternative method, followed by Card et al. (2008), is to split the sample into a test sample, which one can use to select
between multiple potential thresholds, and an analysis sample where the selected threshold can be used. This allows conventional
testing approaches with more standard distributions, but at the cost of reduced test power.
21
Table 2: Testing for a threshold effect in share speaking English
Dependent Variable: Proportion of people in the cohort that speak English
Standard Errors in Parentheses
Cohorts defined by Nationality, County, Grouped Year of Arrival, Grouped Tenure
(1) Unconditional (2) With Fixed Effects (3) With Slopes
Optimal threshold (β1|τ=τ∗) -.509*** -.257*** -.375***
(.025) (.017) (.062)
Constant (β0) .885*** .878*** .913***
(.004) (.002) (.004)
Optimal Threshold Level (τ∗) .31 .31 .31
F-statistic 423 241 37
1% critical value for F-statistic 6.6 6.6 6.6
Cohort Fixed Effects No Yes Yes
Slopes in Nationality Share No No Yes
Observations 1272 1272 1272
Notes. *** denotes significance at 0.1%, ** at 1%, and * at 5% level, when treated as a standalone regression. The
outcome measures the proportion of the cohort that speaks English. Cohorts with English, Canadian, and Irish
nationalities are excluded from the sample, since English is likely to be their mother tongue. Minimum cohort size
of 30. Cohort fixed effects are composed of nationality, arrival year (grouped), and tenure (grouped) fixed effects.
Column (1) is a regression of share speaking English on a constant, and a dummy for whether nationality share in the
county exceeds a threshold. Column (2) allows for cohort fixed effects. Column (3) allows the share speaking English
to also have a slope in nationality share, and allows this slope to vary either side of the threshold. All covariates
except the threshold are demeaned, so the constant (β0) can be interpreted as an estimate of the mean share speaking
English at nationality shares below the threshold. These specifications are run sequentially at different values of the
threshold, varying the threshold between .20 and .40, at intervals of .01. We provide the results for the threshold
among these which produced a regression with the highest F-statistic when tested against the null of no threshold.
The value for this threshold, the estimated F-statistic, and the 1% critical value for this statistic (which corrects for
the repeated testing, see Andrews, 1993) are provided at the bottom of the table.
that report speaking English for a two percentage point range of nationality share. That is,
the leftmost dot shows the mean proportion of immigrants speaking English for all immigrants
in groups that constitute 0-2% of their community. We also plot a kernel-weighted local mean,
fitting it separately either side of a nationality share, zdo, of .31 (below we describe how this
point was chosen). From this figure we can immediately see three things: (i) when immigrants
make up only a small share of the population, the proportion of the cohort that speaks English
is high, at around 90%; (ii) there is a sharp drop in this mean, to less than 40%, when the
immigrant group reaches approximately 1/3 of the population; (iii) there is more variation in
the mean after the threshold. These three findings are consistent with our model. The first
result shows clear evidence that small groups of migrants are assimilated into American culture,
almost all learning the language. The visually striking threshold is clearly suggestive of the type
of strategic complementarity notion at the heart of our model. And the final result is consistent
with the existence of multiple equilibria above the threshold, so that in some communities we
may see assimilation outcomes even at these high nationality shares.
The choice of .31 for the break in our local mean plot comes from performing a formal test
for the presence and location of a significant break. We use the QLR test described above,
searching a grid between [.20, .40] with increments of .01, and test for the most likely value of
the threshold, τ , in a sequence of increasingly flexible models. We consistently find a significant
break in English acquisition, and find the most likely value for the threshold is when the
22
immigrant group constitutes 31% of the population. Table 2 shows the results of the tests.
The first specification includes only a constant and a threshold term, so that any systematic
variation in the share of the cohort speaking English can only be picked up as some kind of
threshold effect. This amounts to imposing that the slope terms are zero (γ0 = γ1 = 0) and
ruling out nationality, year of arrival, and tenure fixed effects (κd = λa = νt = 0) in Equation 3.
We find a strong support for the presence of a threshold, with an F-statistic of 423 (compared
with a 1% critical value of 6.6, taken from Andrews, 1993), and the most likely location for the
threshold is τ ∗ = .31.53
Our next specification allows for nationality, arrival year, and tenure fixed effects, so we only
impose γ0 = γ1 = 0. This allows for the possibility that, for example, Germans might live in
groups with systematically smaller nationality shares than Italians and also find it easier to
learn English, although even this sort of story is unlikely to give rise to so clear a threshold
as seen in Figure 5. Similarly we now allow for variation in settlement patterns and language
acquisition by groups arriving in different years, and who have been present in the US for
different tenures. Again we can strongly reject that no threshold exists (F-statistic of 241),
with the same location. We also now see that the point estimate for β1|τ = τ ∗ is reduced,
showing that some of the reduction in English speaking is explained by the fixed effects.
Finally, in Column (3) we allow also for slopes either side of the potential threshold. Al-
though our model as written does not have such effects, simple extensions of the model, such
as allowing some heterogeneity in costs or benefits, might allow some sort of slope either side of
the threshold. Even allowing for such effects, we find continued support for a threshold, again
when immigrants make up 31% of the local population, with an F-statistic of 37.
Threshold in proportion in-married
We now repeat the above analysis using proportion in-married as the outcome variable. Figure
6 is constructed in the same way as Figure 5, and tells a similar story to that seen with speaking
English. We see that: (i) when immigrants make up only a small share of the population, the
assimilation type outcome is apparent, with relatively less in-marriage; (ii) in-marriage rises
sharply after the threshold; and (iii) we see increased variation after the threshold. The picture
is less sharp here, since the choice of whether to in-marry only captures a single, limited dimen-
sion of interaction between different groups. Additionally, since the timing of the marriages
are unknown, some of the in-marriage might reflect relationships formed prior to immigration,
giving a relatively high level of in-marriage even when nationality shares are very low. Despite
these limitations, at the threshold we see a jump of one-fifth, from less than 60% to more than
75% in-married.
Table 3 shows our main formal results on in-marriage, again testing for the presence and
location of a significant break using the QLR procedure described above (searching a grid
between [.20, .40] with increments of .01). Column (1) estimates our empirical specification
without fixed effects (κd = λa = νt = 0) or slopes (γ0 = γ1 = 0). We find a significant
break (F-statistic of 13.0 against a 1% critical value of 6.6) in in-marriage when the immigrant
group constitutes 35% of the population. Levels of in-marriage rise from 60% to 75% at this
53These results are robust to minimum cohort size chosen. Table C2 replicates Column (1) using different sample size thresholds,
and also using the weighting method discussed earlier. The location and significance of the threshold τ∗ remains unchanged. The
same threshold is also found when we use the completely non-parametric search procedure suggested by Henderson et al. (2015).
23
Table 3: Testing for a threshold effect on share in-married
Dependent Variable: Proportion of the married people in the cohort that are married to
someone of the same nationality
Standard Errors in Parentheses
Cohorts defined by Nationality, County, Grouped Year of Arrival, Grouped Tenure
(1) Unconditional (2) With Fixed Effects (3) With Slopes
Optimal threshold (β1|τ=τ∗) .151*** .065*** .166***
(.042) (.017) (.049)
Constant (β0) .606*** .606*** .562***
(.006) (.003) (.006)
Optimal Threshold Level (τ∗) .35 .23 .23
F-statistic 13.0 15.0 11.5
1% critical value for F-statistic 6.6 6.6 6.6
Cohort Fixed Effects No Yes Yes
Slopes in Nationality Share No No Yes
Observations 1746 1746 1746
Notes. *** denotes significance at 0.1%, ** at 1%, and * at 5% level, when treated as a standalone regression. The
outcome measures the married proportion of the cohort that is ‘in-married’ i.e. married to someone of the same
nationality. Minimum cohort size of 30. Cohort fixed effects are composed of nationality, arrival year (grouped), and
tenure (grouped) fixed effects.
Column (1) is a regression of share in-married on a constant, and a dummy for whether nationality share in the
county exceeds a threshold. Column (2) allows for cohort fixed effects. Column (3) allows the share in-married to
also have a slope in nationality share, and allows this slope to vary either side of the threshold. All covariates except
the threshold are demeaned, so the constant (β0) can be interpreted as an estimate of the mean share in-married at
nationality shares below the threshold. These specifications are run sequentially at different values of the threshold,
varying the threshold between .20 and .40, at intervals of .01. We provide the results for the threshold among these
which produced a regression with the highest F-statistic when tested against the null of no threshold. The value for
this threshold, the estimated F-statistic, and the 1% critical value for this statistic (which corrects for the repeated
testing, see Andrews, 1993) are provided at the bottom of the table.
threshold.54 Column (2) includes fixed effects, and Column (3) additionally allows slopes to
vary. We continue to find a significant break in in-marriage (F-statistics of 15.0 and 11.5
respectively), although it now occurs earlier, when the immigrant group constitutes 23% of the
population.
In summary, we show evidence of stable levels of in-marriage in communities where the
immigrant group is below approximately a quarter of the population (once we account for fixed
effects). When the immigrant group reaches this nationality share we find a significant jump
upwards in rates of in-marriage. This discontinuous increase in segregation, even when using a
partial measure of interaction, provides strongly supportive evidence of the mechanisms driving
our model.
Higher thresholds when culturally closer
Our model predicts not only the presence of a threshold, but also that the threshold should vary
with c. In Table 4 we now estimate the location and significance of the threshold separately
for groups who are from countries with a language ‘near to’ or ‘far from’ English. Splitting the
54These results are robust to minimum cohort size chosen. Table C3 in Online Appendix C replicates Column (1) using different
sample size thresholds, and also using a weighting method, and significance of the threshold τ∗ remains unchanged. The same
threshold is also found when we use the completely non-parametric search procedure suggested by Henderson et al. (2015).
24
Table 4: Comparing the Threshold Locations for Linguistically Far and Near Cohorts
Dependent Variable: (A) Proportion of people in the cohort that speak English; (B) Proportion of cohort
in-married
Standard Errors in Parentheses
Cohorts defined by Nationality, County, Grouped Year of Arrival, Grouped Tenure
(A) Speaking English (B) In-Married
(1) Ling. Far (2) Ling. Near (1) Ling. Far (2) Ling. Near
Optimal threshold (β1|τ=τ∗) -.118*** -.115** -.031 .182*
(.029) (.035) (.034) (.079)
Constant (β0) .876*** .977*** .761*** .402***
(.005) (.007) (.006) (.011)
Optimal Threshold Level (τ∗) .31 .39 .39 .23
F-statistic 17.0 10.7 .80 5.3
5% critical value for F-statistic 3.8 3.8 3.8 3.8
1% critical value for F-statistic 6.6 6.6 6.6 6.6
Cohort Fixed Effects Yes Yes Yes Yes
Slopes in Nationality Share Yes Yes Yes Yes
Observations 810 462 810 936
Notes. *** denotes significance at 0.1%, ** at 1%, and * at 5% level, when treated as a standalone regression. In Panel A the
outcome measures the proportion of the cohort that speaks English, and cohorts with English, Canadian, and Irish nationalities
are excluded from the sample. In Panel B the outcome measures the married proportion of the cohort that is ‘in-married’ i.e.
married to someone of the same nationality. Minimum cohort size of 30. Cohort fixed effects are composed of nationality, arrival
year (grouped), and tenure (grouped) fixed effects.
All columns present regressions of the outcome variable on a constant, an indicator for whether nationality share is above a
threshold, cohort fixed effects, and nationality share itself (allowing for different slopes either side of the threshold). In both
panels, Column (1) includes only the subset of cohorts from our main sample that come from countries whose main language
is deemed far from English (less than 40% mutual intelligibility), whilst Column (2) uses those who are linguistically near. All
covariates except the threshold are demeaned, so the constant (β0) can be interpreted as an estimate of the mean share speaking
English at nationality shares below the threshold. In each case we vary the threshold between .20 and .40, at intervals of .01. We
provide the results for the threshold among these which produced a regression with the highest F-statistic when tested against
the null of no threshold. The value for this threshold, the estimated F-statistic, and the 5% and 1% critical values for this
statistic (Andrews, 1993) are provided at the bottom of the table.
sample into these two cases, we estimate the unrestricted version of Equation 3, analagous to
the third columns of Tables 2 and 3.
For speaking English (Panel A) we find a significant threshold at the 1% level for both far
and near nationalities. As predicted, for near nationalities the threshold occurs at a larger
nationality share (τ ∗ = .39) than for far nationalities (τ ∗ = .31). This is consistent with
our theoretical finding that nationalities that are culturally closer to English need to make up
a relatively larger proportion of the local population before they choose to retain their own
language.
For in-marriage (Panel B), we see a significant threshold still for the near group, at 23%
(significant at 5% level), but we do not find a significant threshold for the far group. In part
this is likely due again to our measure of interactions being only a partial one.
5.4 Discussion of results and limitations
In interpreting our results, we have so far abstracted from the important issue of location choice
and the influence that selection might have on our results. We argue that the presence of such
25
selection does not change our ability to draw conclusions about the key trade-offs inherent in
our model.
The first question of selection is whether migrants choose where to live based on their
individual costs and benefits of interaction, and of the actions they take. The worry might be
that any heterogeneity in the cost of switching cultural practices (for example, different costs
of learning English or different preferences for maintaining religious practices) will manifest as
sorting into different areas. Immigrants with higher costs of switching cultural practices would
have a higher relative return from locating in areas where their group continues to maintain
their own practices. Such selection would only be a problem for us if we were to imagine that
there were discontinuities in the distribution of heterogeneity. Without this, the discontinuity
we observe in behavior must be driven by the structure of the game played by individuals in
these communities. Sorting by individuals may still mean that migrants with different costs
choose to locate in systematically different communities but, as argued by Lazear (1999), this
is simply ‘a question of timing’. If individuals sort, knowing that after choosing their location
their action choice and payoff will depend on those around them, then this is still supportive
of the mechanism of our model. The policy implications will differ, however, if some of the
difference between the assimilated and segregated communities comes from differences in the
costs of assimilation.
The second role for selection relates to who stays in the US. We know from Bandiera et al.
(2013) that there was significant churn, with 60-75% of migrants leaving the US. If migrants
knew ex ante that they were planning to leave the US and go back to their native country,
then they would discount the benefits of adopting the practices of the new community.55 The
benefits would now only be felt for a more limited period, whilst the costs of adoption would
remain unchanged. This is equivalent to these individuals having the same benefits but higher
costs of learning English, and hence the same logic as selection into location applies.
Clearly we do not capture all the richness of the theoretical model. For example, it may
be that multiculturalism manifests as the whole community speaking English to one another
and sharing some other norms of behavior, but at the same time different immigrant groups
maintaining different religious practices, different cuisines, or other different activities. By
looking only at the choice to learn English or not we cannot discern such an outcome. Similarly
we may miss evidence of ‘extreme segregation’ because we do not have a rich enough set of
outcome variables. An extension of the empirical analysis that looks at choices along more
dimensions might allow us to better ‘pull apart’ the different equilibria. However, the data
requirements for such an analysis are strong, including information on social interaction and
various action choices among multiple heterogenous populations. Although we can only pick up
some of the detail of the theoretical model in our empirical analysis, the variables we do consider
show clear evidence that the shape of the relationship is consistent with our predictions.
6 Discussion and Concluding Remarks
This paper builds a tractable framework to answer two questions: what social structures can
arise in heterogeneous populations, and what environmental features determine which social
structure arises. At the center of our analysis is the idea that group behavior and practices,
55If the individuals do not know that they might leave, then it cannot influence their decision.
26
as well as social cohesion, are endogenous. Heterogeneity itself, as well as social relations in
heterogeneous populations, greatly depend on the environment.
Understanding how the environment shapes heterogeneous populations is critical for policy
design. Integration of immigrant and minority groups receives a lot of political attention, and
governments frequently propose and implement policies intended to influence the way minority
groups integrate. Our findings paint a nuanced picture of both the relevance and effects of such
policies. We briefly consider what our framework implies for two important policies.
First, our model suggests that secularization policies, rather than restraining religion, can
actually make groups more likely to maintain diverse religious practices. Secularism prescribes
some degree of restriction on religion in public life. In France in 2004 the principle of secularism
was applied to ban all conspicuous religious symbols (including veils of any kind) from public
schools. Our framework shows that policies which reduce the relevance of cultural practices in
everyday life, such as removing religion from state institutions, can enable multiculturalism.
That is, it implies that maintaining different religions is easier in a more secular society!
Second, our framework implies that policies which reduce barriers to interaction across
groups (e.g. school bussing or desegregation) may result in a response whereby groups not
only remain segregated but amplify their differences. Reduced costs of interaction require
minority groups to distinguish themselves even more if they are to remain segregated. Thus
increased opportunities for interaction can lead to ‘secondary’ differences where minority groups
begin to create new distinguishing features and emphasise previously insignificant behaviors as
culturally important.
Our framework does not rule out a role for policy however. The key welfare tension in
the model comes from the inability of group members to coordinate, which can lead to ‘in-
efficient segregation’.56 A failure to coordinate among immigrants allows segregation to arise
in situations where a collective move to assimilation or multiculturalism would be a Pareto
improvement. This is because when groups are segregated under Nash equilibrium, a minority
individual does worse by switching culture and joining the majority group, even though the mi-
nority group as a whole might do better by adopting majority culture. Policy can influence this
by reducing the costs of switching culture for at least some minority individuals, thus breaking
down the segregation equilibrium. Schooling policy provides an example of this, where the
teaching of national values to the children of immigrants makes it easier for them to integrate
with natives, thus making it harder for their parents to remain segregated (see Alesina and
Reich (2015), Bandiera et al. (2015) and references therein).
Of course, the costs of switching culture and the ubiquity of cultural and religious practices
in daily life could be influenced not only by governments by also by individuals and groups.
‘Group leaders’ might be able to influence these parameters to achieve their individual aims. For
example religious leaders might care most about preserving religious adherence, and attempt to
adjust the cost of switching practices or increase the ubiquity of religious activities, even at the
cost of lowering the utility of group members. Alternatively, the costs of switching culture and
the relevance of cultural and religious practices could emerge endogenously in a decentralized
model. This paper takes a key step in understanding endogeneity of heterogeneity.57 However,
there remain many important avenues for further work in this direction.
56See Online Appendix B for an expanded welfare analysis.57For work on the endogeneity of national culture see Alesina and Reich (2015).
27
A second important direction for future research would be to allow for not just more than two
types of individual but different dimensions of types. For example, we may have two different
nationality groups with ex ante different practices, but each nationality group may itself be
composed of a mix of people of two different religions which have their own associated practices.
An illustration is provided by the Wars of the Three Kingdoms, which saw Protestants in
England, Scotland, and Ireland (before the formation of the United Kingdom) fighting together
against Catholics from their own countries. Adapting our framework with such an addition may
provide answers to questions about when and why one divide (for example religion) becomes
more important than another (for example ethnicity) in different societies.
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Appendix A
Proof of Proposition 1
In a Nash equilibrium all players of the same type must play the same action set. Suppose
not, and that two players of the same type, i and j, play different action sets. One player,
without loss of generality player i, must do weakly worse than the other. Then i can do strictly
better by mimicking j’s strategy and also forming a tie with j. Forming a tie with j ensures
that player i either has an additional tie compared to j’s previous strategy, or that the value i
receives from the tie between himself and j is higher than the value j would have been receiving
from this tie, since they now coordinate on more actions. Hence in equilibrium i and j must
play the same actions. In equilibrium each player must also link with all others who adopt the
same action set since the value of such a tie is strictly positive. Thus all individuals in the same
group form a tie in equilibrium.
Suppose 1−α < L. Suppose there is (at least) one tie between different types in equilibrium.
If the two types do not have culture in common then the payoff from this tie is at most
1−α−L < 0, and the individual can do strictly better by dropping the tie. In equilibrium the
two groups must have culture in common for there to be a tie between groups. Suppose the two
groups have culture but do not have non-cultural practices in common. Then a player from the
weakly smaller group can do strictly better by adopting the non-cultural action of the weakly
larger group and linking accordingly. Therefore, if there is a tie between groups in equilibrium,
all individuals must play the same action set and thus all individuals must be linked.
Suppose there are no ties between different types in equilibrium. If the two types adopt
the same cultural practices but do not have non-cultural practices in common, then a player
from the weakly smaller group can do strictly better by adopting the non-cultural action of
the weakly larger group and instead linking with that group. If the two types adopt the same
31
action set, then they would have strictly higher utility by linking across groups. Thus if there
are no ties between different types, in equilibrium the two types must adopt different cultural
actions. Further, each type must adopt its own cultural action since, if not, a type M must
be doing weakly better by playing xm and linking with his group than playing xM and linking
with the other group. But then a type m must do strictly better by playing xm, linking with
the other group and not paying the cost of switching culture. A contradiction.
We have ruled out all states apart from assimilation and segregation states as possible
equilibria. Assimilation is an equilibrium under all parameter values since we assumed (1 −L)(n − 1) − c > max{0, (1 − α − L)(n − 1)}, and so all deviations by an individual do worse.
Segregation is an equilibrium if (1 − L)(nm − 1) ≥ (1 − L)(n − nm) − c. It is straightforward
to see that under this condition all deviations by an individual do worse.
Suppose 1− α ≥ L. Suppose there are no ties between different types in equilibrium. If the
two types adopt the same cultural practices but do not have non-cultural practices in common,
then a player from the weakly smaller group can do strictly better by adopting the non-cultural
action of the weakly larger group and instead linking with that group. If the two types adopt
the same action set, then they would have strictly higher utility by linking across groups. The
remaining possibility is that the two types adopt different cultural actions. If the two types
adopt the same non-cultural action the value of a tie with the other type is 1−α−L ≥ 0 and so
this is not an equilibrium. Thus different types must adopt different cultural and non-cultural
actions. Each type must also adopt their own cultural action, by the above.
Suppose there is (at least) one tie between different types in equilibrium. If the two types
have cultural practices in common then, by the proof above, they must also have non-cultural
actions in common and thus all individuals must be linked. If the two types do not have
cultural practices in common, then suppose they also do not have non-cultural practices in
common. Then the value of a tie between types is strictly negative and the individual with a
tie to a different type does strictly better by dropping it. If the two types do not have cultural
practices in common, then the remaining possibility is that they do have non-cultural practices
in common. Then the value of a tie with the other type is weakly positive and all individuals
must be linked.
We have ruled out all states apart from assimilation, segregation, and multiculturalism
as possible equilibria. Assimilation is an equilibrium under all parameter values. Given the
assumption (1−L)(n− 1)− c > (1− α−L)(n− 1), any deviation from assimilation by a lone
individual results in a strictly lower payoff.
Segregation is an equilibrium when no player wants to deviate. No player wishes to deviate,
play the action set of the other group, and link only with the other group if and only if
(1− L)(nm − 1) ≥ (1− L)(n− nm)− c
or deviate, switch non-cultural action, and link with the other group as well as their own type
if and only if
(1− L)(nm − 1) ≥ (α− L)(nm − 1) + (1− α− L)(n− nm).
It is straightforward to see that under these conditions all other deviations by an individual
away from segregation do worse. Multiculturalism is an equilibrium when no player wishes to
32
mimic the strategy of the other group if and only if
(1− L)(nm − 1) + (1− α− L)(n− nm) ≥ (1− L)(n− nm) + (1− α− L)(nm − 1)− c.
It is straightforward to see that under these conditions all deviations by an individual do worse.
Proof of Proposition 2
The process described defines an aperiodic and irreducible Markov Chain; it therefore has a
unique stationary distribution. We briefly describe our notation and method of computing the
stationary distribution as ε → 0. We refer to the strict Nash equilibria of the game, denoted
here by σ, as nodes. There is a minimum number of players required to make a mistake in their
strategy for there to be a positive probability of transiting from one strict Nash equilibrium,
call it σ′, to another strict Nash equilibrium, say σ, by the best response dynamics. We refer
to this minimum mistake transition from node σ′ to node σ as the link from σ′ to σ and the
minimum number of players required to make a mistake as its weight. A path from node σ′ to
σ is a sequence of directed links starting at σ′ and connecting to σ. A σ − tree is graph with
no cycles such that for each strict Nash equilibrium σ′ 6= σ a unique path connects it (directly
or indirectly) to σ. The stochastic potential of σ is the total weight of the σ− tree whose links
sum to the lowest total weight. A stochastically stable state is a state with positive probability
in the stationary distribution when ε→ 0. The stochastically stable states are the strict Nash
equilibria with minimum stochastic potential. This result and details can be found in Young
(1993).58
To find the σ − tree for each strict Nash equilibrium σ would involve enumerating a lot of
possibilities. We use a tree-pruning argument to simplify things. The proof here is given for
the parameter range 1 − α < L. The result for the reverse inequality is similar and given in
Online Appendix B.
Lemma 1 Partition the strict Nash equilibria into three sets: segregation; assimilation to M
(‘M-assimilation’); and assimilation to m (‘m-assimilation’). Such a set is denoted Σx, and
x ∈ {s,M,m} respectively indexes the set. All equilibria in a set, σ ∈ Σx, have the same
stochastic potential.
Proof. Equilibria in a given set are symmetric so, for any σ − tree, an equivalent σ − tree can
be constructed with the same total weight for any other equilibrium in the same set.
Lemma 2 A strict Nash equilibrium σ ∈ Σx has lower stochastic potential than a strict Nash
equilibrium σ ∈ Σy, y 6= x, if the link from σ to σ has lower weight than the link from σ to σ
for all σ ∈ Σz, for all z 6= x.
Proof. Take a tree which defines the stochastic potential of node σ. Denote by pσ1σ the path
from some node denoted σ1 to σ in such a tree. Next find the link from σ to σ and denote
this link by lσσ. Adding this link to the tree forms a graph that contains paths from all nodes
to σ, since the graph contains the path pσ1σ + lσσ for all σ1. Observe that the tree to σ must
involve a link from the chosen σ equilibrium to some other node σ2, denoted lσσ2 . This link,
lσσ2 is incorporated into a path from σ1 as follows: pσ1σ + lσσ2 + pσ2σ + lσσ (where it can be that
58This result also requires that the process without experimenting converges almost surely to a strict Nash equilibria. This is
straightforward and, for completeness, is found in Online Appendix B.
33
σ1 = σ, so the path starts at σ, and/or σ2 = σ so the link from σ in the original tree is directly
to σ). It is clear that by deleting the link lσσ2 there is still a path from any node to the node
σ. If the weight of the link added (lσσ) is lower than the weight of the link deleted (some lσσ2)
then the lowest weight tree to σ has lower total weight than the lowest weight tree to σ.
To show that one need only examine links from σ out of the set Σx, observe the following.
If the link lσσ2 is to some σ2 ∈ Σx, then the path pσ2σ must include a link from a node in Σx,
denoted σ3 to a node, denoted σ4, where σ4 ∈ Σy, y 6= x, (where it could be that σ3 = σ2). The
path lσσ2 + pσ2σ must include a path from σ to σ3 that links only nodes in Σx. Now delete the
link from σ3 to σ4 ∈ Σy, lσ3σ4 , and reverse all links on the path from σ to σ3. This changes the
path pσ1σ + lσσ2 + pσ2σ + lσσ into the following paths pσ1σ, pσ3σ, and pσ4σ + lσσ. By deleting the
link lσ3σ4 there is still a path from any node on the path pσ1σ + lσσ2 + pσ2σ + lσσ to the node
σ. Similarly if the path from any node j includes any node on the path pσ1σ + lσσ2 + pσ2σ + lσσthen there continues to be a link to σ. Observe that reversing a link between two nodes in a
given set does not change the weight of the link. Thus if the weight of link lσ3σ4 is higher than
that of link lσσ then the lowest weight tree to σ has lower total weight than the lowest weight
tree to σ. To conclude the proof, note that equilibria are symmetric so the path lσ3σ4 will have
an analogous path from σ. �
To simplify notation, we denote an equilibrium where both types play (xM , yA) by (M,A)
and analogously for other assimilation equilibria. We denote an equilibrium where type M
play (xM , yA) and type m play (xm, yB) by (M,A;m,B), and analogously for other segregation
equilibria. The lowest weight link between each group, denoted φ, is detailed in Table A1. For
now we allow the weight of a link to be a real number, φ ∈ R, and address the integer nature
of mistakes at the end of the proof. Given Lemmas 1 and 2, the following statements follow in
a straightforward manner from Table A1.
M-assimilation has lower stochastic potential than segregation when φsM < φMs.
Take the lowest weight tree to any segregation strict Nash equilibrium. The lowest weight link
from this node to some M -assimilation equilibrium is φsM . It can be seen from Table A1 that
the weight of any link from M -assimilation to m-assimilation, or to segregation, is at least φMs.
Thus the result follows from Lemmas 1 and 2.
Segregation has lower stochastic potential than M-assimilation when φMs < φsM .
Take the lowest weight tree to any M -assimilation equilibrium. The lowest weight link from
this node to some segregation equilibrium is φMs. Any link from segregation to M -assimilation,
or to m-assimilation, has weight at least φsM . By Lemma 1 and 2 the statement holds.
Segregation has lower stochastic potential than m-assimilation when φMs < φsM .
The lowest weight link from any m-assimilation equilibrium to segregation is φMs. The result
follows as above.
M-assimilation has equal stochastic potential to m-assimilation when φsM < φsm < φMs.
Take the lowest weight tree to any M -assimilation equilibrium. The lowest weight link from this
node to some m-assimilation equilibrium is φMs, since φsm < φMs. Any link from m-assimilation
34
to M -assimilation, or to segregation, has weight at least φMs, since φsM < φMs. By Lemma
1 and Lemma 2, m-assimilation has weakly lower stochastic potential than M -assimilation.
Starting from the lowest weight tree to any m-assimilation equilibrium repeat the process to
show M -assimilation has weakly lower stochastic potential than m-assimilation.
M-assimilation has lower stochastic potential than m-assimilation when φsM < φMs < φsm.
Take the lowest weight tree to any m-assimilation equilibrium. Suppose this lowest weight tree
includes a link from one of the segregation equilibria to (m,A) or (m,B). The minimum weight
link from some segregation equilibrium to some m-assimilation equilibrium is φsm. Delete this
link and form a link from the same segregation equilibrium, by the lowest weight link, to either
(M,A) or (M,B), which will have weight φsM . The new graph has strictly lower total weight.
Now form a link from the m-assimilation equilibrium to the same node, either (M,A) or (M,B),
which will be weight φMs, and delete a link from the node either (M,A) or (M,B), which are
all of weight at least φMs. Thus we have a graph from all nodes to (M,A) or (M,B) with
strictly lower total weight than the tree to m-assimilation.
Suppose instead the lowest weight tree to any m-assimilation equilibrium has no link from
a segregation equilibria to (m,A) or (m,B). Then there must be a link from either (M,A) or
(M,B) to either (m,A) or (m,B) in order to have a tree to m-assimilation. Now a link from
either (M,A) or (M,B) has the same weight to either (m,A) or (m,B) so we can replace it
with a link to the chosen m-assimilation equilibrium while maintaining the tree and without
affecting the weight of the tree. Now we delete this link and form the reverse link from the
chosen m-assimilation equilibrium to the node (M,A) or (M,B) whose link to m-assimilation
was just deleted. The graph now has strictly lower weight and maintains a link to all nodes.
Above we suppose φ ∈ R. Since our population is discrete, the minimum weights are in
fact the lowest integer greater than or equal to a given φ. Given the discrete nature of the
population the equilibrium is guaranteed to be unique when φMs < φsM if φsM − φMs ≥ 1
and when φMs > φsM if φMs − φsM ≥ 1. Since φsM R φMs implies nm R n − c/(1 − L),
φsM − φMs ≥ 1 is equivalent to nm ≥ n − c/(1 − L) + 1 = δ∗ + 1, and φMs − φsM ≥ 1 is
equivalent to nm ≤ n− c/(1− L)− 1 = δ∗ − 1. M -assimilation has lower stochastic potential
than m-assimilation when φsm > φMs, allowing for discreteness requires φsm − φMs ≥ 1 which
implies nm ≤ c/(1− L)− 1. This final inequality is Assumption 2.
35
Tab
leA
1:T
he
low
est
wei
ght
lin
kfr
om
an
od
ein
on
ese
tto
an
od
ein
ad
iffer
ent
set,
for
1−α<L
an
dαRL
.
Set
sli
nke
dM
inim
um
nu
mb
erof
mis
takes
totr
an
sit
Note
sW
eight
bet
wee
nse
tsis
min
.φ
sati
sfyin
g:
den
ote
d
segr
egat
ion
to(1−L
)(nM
+φ
)−c≥
(1−L
)(nm−φ−
1)
e.g.
from
(M,A
;m,B
)to
(M,A
).φsM
M-a
ssim
ilat
ion
Th
isin
volv
esty
pem
makin
ga
mis
take
of
(M,A
).
segr
egat
ion
to(1−L
)(nm
+φ
)−c≥
(1−L
)(nM−φ−
1)
e.g.
from
(M,A
;m,B
)to
(m,B
).φsm
m-a
ssim
ilat
ion
Th
isin
volv
esm
ista
kes
of
(m,B
).
M-a
ssim
ilat
ion
(1−L
)φ≥
(1−L
)(n−φ−
1)−c
e.g.
from
(M,A
)to
(M,A
;m,B
).φMs
tose
greg
atio
nT
his
invo
lves
mis
takes
of
(m,B
).
m-a
ssim
ilat
ion
(1−L
)φ≥
(1−L
)(n−φ−
1)−c
e.g.
from
(m,B
)to
(M,A
;m,B
).φMs
tose
greg
atio
nT
his
invo
lves
mis
takes
of
(M,A
).
m-a
ssim
ilat
ion
(1−L
)φ≥
(1−L
)(n−k−
1)−c
e.g.
from
(m,A
)to
(M,A
).m
ax{φ
Ms,φsM}
toM
-ass
imil
atio
nan
dT
yp
eM
want
totr
an
siti
on
aft
erat
leastφ
mis
take
sof
(M,A
).
(1−L
)(nM
+φ′ )−c≥
(1−L
)(nm−φ′−
1)
On
ceall
typ
eM
hav
em
oved
,ty
pem
want
totr
an
siti
on
ifat
leastφ′
mis
take
sare
mad
e(b
yty
pem
).
Th
em
inim
um
nu
mb
erof
mis
take
sm
ust
sati
sfy
both
ineq
uali
ties
.
M-a
ssim
ilat
ion
(1−L
)φ≥
(1−L
)(n−k−
1)−c
e.g.
from
(M,A
)to
(m,A
).m
ax{φ
Ms,φsm}
tom
-ass
imil
atio
nan
dT
yp
em
want
totr
an
siti
on
aft
erat
leastφ
mis
take
sof
(m,A
).
(1−L
)(nm
+φ′ )−c≥
(1−L
)(nM−φ′−
1)
On
ceall
typ
em
hav
em
oved
,ty
peM
want
totr
an
siti
on
ifat
leastφ′
mis
take
sare
mad
e(b
yty
peM
).
36
Online Appendix B Additional discussion of the model and proofs
Extending the model to action choices along more than two dimensions
We have modeled action choice along two dimensions. In reality, individuals make many more
than two decisions. The minority group may initially speak one language and the majority group
another, the minority group may participate in certain religious practices and the majority
group others, the culture of the minority group may permit alcohol while that of the majority
group forbids it, and so on. The baseline model condenses these many choices into just two: a
choice of cultural action, which represents choice over practices for which there is type-specific
cost, and a choice of non-cultural action, which represents choice over practices which have
no type specific cost. Very little intuition is lost through this abstraction. Here we relax this
assumption and highlight the novel features which arise.
Let xi1 denote i’s choice of language, which is chosen from the set {xm1 , xM1 }, where xm1 is
the language of the minority group and xM1 is the language of the majority. If two individuals
speak the same language then the benefit from interaction increases by α1. Suppose the cost
to switching language is c1. Let xi2 denote i’s choice to drink alcohol or not, chosen from
the set {xm2 , xM2 }, where group m permits alcohol and group M does not. Suppose α2 is the
importance of having this in common in terms of enabling interaction and economic exchange.
Let c2 be the cost of switching. Generally let (xi1, . . . , xiQ) denote the Q ‘cultural choices’ of
individual i. The set (xm1 , . . . , xmQ) contains the ex ante cultural practices of group m, and the
set (xM1 , . . . , xMQ ) the ex ante cultural practices of group M . Some cultural practices may be
very costly to switch away from and others less so, represented by the cost of switching for each
practice, cr, r ∈ {1, . . . , Q}. Some practices may be very relevant to interaction and others less
so, represented by αr, r ∈ {1, . . . , Q}. Non-cultural actions are modeled similarly. Individual
i chooses R non-cultural actions, (yiQ+1, . . . , yiQ+R), where for each action there is a choice,
{yAQ+1, yBQ+1}, . . . , {yAQ+R, y
BQ+R}, where the choice is denoted by the superscripts A and B. Any
non-cultural choice has an associated cost of zero, cr = 0. The importance of each practice in
terms of social interaction is given by αr, r ∈ {Q+ 1, . . . , Q+R}. We normalize these to sum
to one.Q+R∑r=1
αr = 1.
Each individual chooses social ties as before, but now chooses a larger set of actions
(xi1, . . . , xiQ, yiQ+1, . . . , yiQ+R). The utility function is now
uk(si, s−i) =n∑j=1
∑r∈{1,...,Q}
αrπr(xir, xjr) +∑
r∈{Q+1,...,Q+R}
αrπr(yir, yjr)− L
gij−∑r
ck(xir) (4)
where
π(xir, xjr) =
{1 if xir = xjr0 if xir 6= xjr
π(yir, yjr) =
{1 if yir = yjr0 if yir 6= yjr
37
and for individual i of type k ∈ {M,m}
ck(xir) =
{0 if xir = xkrcr if xir 6= xkr
The Nash equilibria and comparative statics with multiple dimensions are analogous to
those in the main model presented. We highlight only the notable richness that adding more
dimensions brings. Under similar conditions to the main model, assimilation outcomes are
Nash equilibria, where all individuals form a tie, all adopt the same practices where one group
adopts all the cultural practices of the other group. There now exist what we refer to as
‘melting pot’ equilibria, a type of assimilation equilibrium where all individuals form a tie and
all adopt the same practices (both cultural and non-cultural), but the action choices could
be a mix of the ex ante cultural practices of the two groups. Thus the ex post culture that
emerges is a mix of the ex ante different cultures (Kuran and Sandholm, 2008). There are
also multiculturalism equilibria, similar to main model, whereby all individuals form a tie, have
all non-cultural practices in common, but the different groups maintain at least some of their
different cultural practices. There is some added richness here which does not appear when
there are only two action choices: the smaller the minority group the fewer minority traits
they can sustain in a multiculturalism equilibrium. There is a sequence of multiculturalism
equilibria, from the case where each group maintains all their own cultural actions, to fewer
and fewer of one group’s traits being retained, moving towards assimilation. The only other
Nash equilibria are segregation equilibria: individuals form ties only within their group and
not to the other group, and adopt sufficiently different action sets (this includes versions of the
‘extreme segregation’ outcome).59 There is also added richness here: the smaller the minority
group, the more they must differentiate their practices in order for segregation to be a Nash
equilibrium. This is consistent with empirical findings (Bisin et al., 2013) and theoretical
findings (Bisin and Verdier, 2000) that smaller minority groups must exert more effort to retain
their diverse traits.
Discussion of alternative models of link formation
As mentioned, the model of link formation presented here is not the only way to model social
interactions. For example, the Nash equilibria in this paper all satisfy the definition of pairwise
stability, an important measure of stability in network formation (Jackson and Wolinsky, 1996).
In fact, a two-sided link formation model, related to pairwise stability, where an individual could
delete any number of links and form any number of mutually agreed upon links would produce
the same outcomes as the link formation model we use. There are many ways to model how
individuals form ties and in any framework a choice must be made. The key consideration
in choosing the model of link formation here was to avoid a multitude of equilibria (often a
feature of social interaction models) that add complication without giving additional intuition
with respect to the questions we seek to answer.
Another way of modeling social interaction would be to add a cap on the number of ties
an individual can form, or to assume decreasing returns in the number of ties. With such
assumptions, the qualitative results would remain in smaller communities and we would see
59The segregation equilibria have the feature that, although a group will mainly adopt its own cultural practices, it may be that
a group adopts a few cultural practices of the other group. This is a feature of examining Nash equilibria in a coordination game.
38
more segregation in larger populations where the cap on the number of contacts becomes
relevant to decision making.
Welfare results and discussion
We will consider which states are Pareto efficient and which would be chosen by a social planner,
where the social planner maximizes the sum of utilities in the population∑i
uk(si, s−i).
We highlight three findings.
First, multiculturalism is Pareto efficient under the same parameters that multiculturalism
will persist in the long run. Similarly segregation is Pareto efficient exactly when it will persist
in the long run.60 That is, segregation and multiculturalism are likely to emerge in the long-run
exactly when this outcome is in the minority group’s best interest.61
Second, assimilation is always Pareto efficient. A group can do no better than when the
other group assimilates, since this maximizes interaction and the other group pays the cost of
switching culture. This suggests that when a group is able to do so, it will put pressure on
other groups to do the assimilating. Calls for immigrants to adopt local languages and cultural
practices by the public or by politicians are thus unsurprising. In contrast, a social planner will
not always choose assimilation.
The third point to highlight is the result of Proposition 3 (below). This reveals a tension
between when segregation is a Nash equilibrium, when it is optimal for the minority group,
and when it is optimal for the social planner. Proposition 3 says that segregation is a Nash
equilibrium under a larger set of parameters than those for which it is Pareto optimal, and
segregation is Pareto optimal under a larger set of parameters than those for which the social
planner would choose to implement segregation. The reason for this tension is that segregation is
a Nash equilibrium when, given the population is segregated, no individual from the minority
group does better by switching culture and joining the majority group. However, even if a
minority individual in this situation prefers to stick with minority culture, it could be that
the minority group as a whole would do better by adopting majority culture. The social
planner chooses segregation under a yet smaller set of parameters, since the social planner also
incorporates the costs of segregation to the majority group. This same reasoning applies to
multiculturalism under the parameters 1− α ≥ L.
Let δW denote the threshold size of the minority group above which segregation is chosen
by the social planner, let δP denote the threshold above which segregation is Pareto efficient,
and δ is the threshold above which segregation is a Nash equilibrium, given in Proposition 1.
Analogously for ηW , ηP , and η for multiculturalism. Observe in Proposition 3 that a social
planner would choose multiculturalism when 1− α ≥ L and the minority group is large, again
lending support to the proposed ideal of a multicultural society.
Proposition 3 When 1− α < L:
δW > δP > δ.60The only difference is that we do not need to account for discreteness as in Proposition 2.61Excluding the case where the majority group assimilate to minority culture, which the minority group would prefer but cannot
influence.
39
When 1− α ≥ L:
ηW > ηP > η.
Proof of Proposition 3.
Any assimilation state is always Pareto efficient since any individual whose cultural action is
played attains his maximum payoff, doing strictly worse under any other state.
Segregation is Pareto efficient when (5) holds and 1−α < L. Each individual in the minority
group gets a strictly higher payoff from segregation than assimilation to majority culture when
(1− L)(nm − 1) > (1− L)(n− 1)− c
rewritten
nm/n > 1− c/n(1− L). (5)
It is immediate to see that individuals in the minority group get a higher payoff from segre-
gation than any other state where only the majority cultural action is played. Symmetrically
for the majority group. It remains to show that some individual does strictly worse in any
state, other than segregation, where at least two individuals adopt different cultural actions.
Since 1 − α < L, any individual who links with someone adopting a different cultural action
does strictly worse. If fewer individuals play the minority cultural action, then those remaining
playing the minority action must be strictly worse off (symmetrically for the majority cultural
action). The remaining possibility is that the size of each group playing either cultural action
remains the same, but some majority and minority individuals switch strategy. Clearly some
individuals do strictly worse.
Multiculturalism is Pareto efficient when 1 − α ≥ L and the minority group does strictly
worse by assimilating
(1− L)(nm − 1) + (1− α− L)nM > (1− L)(n− 1)− c
rewritten
nm/n > 1− c
αn.
The proof follows from above, also noting that under 1 − α ≥ L individuals do weakly better
by all adopting the same non-cultural action and linking.
Assimilation by the minority gives higher welfare than any other state in which a single
cultural action is played. When 1 − α < L, in any state that maximizes welfare with two
cultural actions being played, any two individuals playing the same cultural action must also
play the same non-cultural action and must link, and any two individuals playing different
cultural actions must not link. Segregation gives weakly higher total welfare than assimilation
by the minority when
nm(1− L)(nm − 1) + nM(1− L)(nM − 1) ≥ nm[(1− L)(n− 1)− c] + nM(1− L)(n− 1)
rewritten
nm/n ≥ 1− c/2n(1− L).
40
It remains to show that total welfare is weakly lower when some other combination of individuals
play different cultural actions. Suppose a smaller group of minority individuals, nm − n1, play
the minority cultural action and the rest, n1, play the majority cultural action, for n1 ∈{1, . . . , nm − 1}. Total welfare is higher than under segregation if
(nm − n1)(1− L)(nm − n1 − 1) + (nM + n1)(1− L)(nM + n1 − 1)− n1c
≥ nm(1− L)(nm − 1) + nM(1− L)(nM − 1)
rewritten
(nM + n1)/n− c/2n(1− L) ≥ nm/n.
A contradiction. The same applies if a smaller group of majority individuals play the majority
cultural action. Finally it is clear that each type playing their respective cultural action maxi-
mizes welfare in this situation.
Multiculturalism maximizes the social planner’s problem when 1−α ≥ L and it gives higher
total welfare than assimilation by the minority, that is
nm(1− L)(nm − 1) + nM(1− L)(nM − 1) + nm(1− α− L)nM
+ nM(1− α− L)nm ≥ nm[(1− L)(n− 1)− c] + nM(1− L)(n− 1)
rewritten
nm/n ≥ 1− c/2αn.
The result follows similarly to above. �
Heterogeneous costs of switching culture.
Suppose for members of the immigrant group that the costs of switching culture are given by
c1m ≤ c2m ≤ . . . ≤ cnmm
and similarly for the native group
c1M ≤ c2M ≤ . . . ≤ cnMM .
Let us consider the case where 1− α < L. Then individuals will not form a tie if they have
only non-cultural actions in common.
Assimilation (to immigrant or native culture) is always an equilibrium provided that the fol-
lowing condition, analogous to the previous no man is an island assumption, holds:
(1− L)(n− 1)− ch > 0 ∀h ∈ {1m, 2m, . . . , nmm, 1M, 2M, . . . , nMM}.
Suppose all individuals in the population adopt the same cultural action, then by the same
argument as the proof of Proposition 1, in equilibrium all must adopt the same non-cultural
action and form a link. Thus assimilation states are Nash equilibria under all parameters and
they are the only Nash equilibria where all individuals adopt the same cultural action.
41
Suppose some individuals in the population adopt different cultural actions. Let n1 be the
size of the group adopting xm and n2 be the size of the group adopting xM . In equilibrium
those playing the same cultural action must play the same non-cultural action and link other-
wise there are profitable deviations.
Suppose n1 ≥ n2. Then any minority individuals playing xM do strictly better by switch-
ing to play xm and so in any Nash equilibrium all minority individuals must be in group n1.
No minority individual wishes to deviate. A majority individual in group n1 with cost ch does
not want to deviate if
(1− L)(n1 − 1)− ch ≥ (1− L)n2
and a majority individual in group n2 with cost c′h does not want to deviate if
(1− L)(n2 − 1) ≥ (1− L)n1 − c′h.
Note that the final two conditions hold only if all majority individuals in group n1 have a lower
cost of switching, ch, than all individuals in group n2.
Instead suppose n2 ≥ n1. In any Nash equilibrium all majority individuals must be in group
n2. No minority individual with cost ch wishes to deviate from group n2 if
(1− L)(n2 − 1)− ch ≥ (1− L)n1
No minority individual with cost c′h wishes to deviate from group n2 if
(1− L)(n1 − 1) ≥ (1− L)n2 − c′h.
This is the same as above, although it is not guaranteed that it will hold even if the whole
minority group is part of n1. As with the homogeneous cost case segregation only holds when
the minority group is large enough.
Additional proofs for Proposition 2
Proof that the Markov process converges almost surely to a strict Nash equilibrium
We show there exists no recurrence class other than the strict Nash equilibria. To do so it
suffices to show that from any state there are a finite number of positive probability events
which lead to a strict Nash equilibrium.
Suppose n1 ≤ n players play (M,A), n2 ≤ n players play (m,A), n3 ≤ n players play (M,B)
and n4 ≤ n players play (m,B), where either type can be playing either action. With positive
probability a given type M is selected to update his strategy. Suppose the type M does (at
least weakly) best by playing (M,A), and chooses (M,A). Suppose next a different individual
of type M is selected to update his strategy; again this occurs with positive probability. This
individual must strictly prefer (M,A) since the payoff from playing (M,A) relative to other
actions has strictly increased and it was weakly optimal for a type M in the previous period.
Let this continue such that all type M are selected and no type m. All type M now play
(M,A). Next a given type m is selected to update his strategy with positive probability. His
optimal strategy may be either (M,A), (m,A), or (m,B). He chooses one of these actions. Let
42
this continue until all type m have been selected to update their strategy. The action chosen
by the first type m agent selected to update his strategy is strictly best for all type m that
follow. The process arrives at a strict Nash equilibrium since all type M play (M,A) and all
type m play one of either (M,A), (m,A), or (m,B). Thus we have either assimilation by type
m, segregation, or multiculturalism. The same argument holds if instead the initial type M
selected to update his strategy selects (M,B). If instead the initial type M selected to update
his strategy selects (m,A), then all type M or m that follow must do strictly best by playing
(m,A). The result follows. Similarly if instead the initial type M selected does best by playing
(m,B).
Proof of Proposition 2 for parameters 1− α > L.
Lemma 1 becomes Lemma 3:
Lemma 3 Partition the strict Nash equilibria into four sets: segregation; multiculturalism; M-
assimilation; and m-assimilation. Such a set is denoted Σx and x ∈ {s, u,M,m} respectively
indexes the set. All equilibria in a set, σ ∈ Σx, have the same stochastic potential.
Lemma 2 continues to apply. Notation is simplified in the same way as the proof for 1−α < L.
Table B1 finds the lowest weight link between sets i.e. from some node in set Σx to some node
in another set Σy.
Multiculturalism has lower stochastic potential than M-assimilation when φMu < φuM .
Take the lowest weight tree to an M -assimilation equilibrium. The lowest weight link from
this node to some multiculturalism equilibrium is φMu. It can be seen from Table B1 that the
weight of any link from multiculturalism to M -assimilation, m-assimilation or segregation is
at least the minimum of φuM and nm−12
+ 1−α−L2(1−L)
nM . Observe φuM + φMu = nm − 1 and so
φMu <nm−1
2. Thus the result follows from Lemmas 2 and 3.
Multiculturalism has lower stochastic potential than m-assimilation when φMu < φuM .
The transit from m-assimilation to multiculturalism also has weight φMu so the argument fol-
lows as above.
M-assimilation has lower stochastic potential than multiculturalism when φuM < φMu.
Observe that φuM is the lowest weight transition from multiculturalism to M -assimilation when
φuM < φMu since then φuM < nm−12
. From Table B1, any transition from M -assimilation to
multiculturalism, segregation or m-assimilation has weight at least φMu (since φMs > φMu).
The result follows from Lemmas 2 and 3.
M-assimilation has lower stochastic potential than segregation when φuM < φMu.
The transit from segregation to M -assimilation has lower weight than φuM and so the result
follows from above.
Multiculturalism has lower stochastic potential than segregation when φMu < φuM .
The inequality φMu < φuM implies n−12− c
2α< nm−nM−1
2+ c
2αwhich implies nM < c
α. Thus
the lowest weight link from segregation to some multiculturalism equilibrium requires nm−12−
43
Tab
leB
1:F
ind
ing
the
low
est
wei
ght
lin
kfr
om
anod
ein
on
ese
tto
an
od
ein
ad
iffer
ent
set,
for
1−α>L
an
dαTL
.
Set
sli
nked
Min
imu
mnu
mb
erof
mis
takes
totr
an
sit
Note
sW
eight
bet
wee
nse
tsis
min
.φ
sati
sfyin
g:
den
ote
d
M-a
ssim
ilat
ion
(1−L
)φ+
(1−α−L
)(n−φ−
1)≥
e.g.
from
(M,A
)to
(MA,m
A).
φMu
tom
ult
icu
ltura
lism
(1−L
)(n−φ−
1)+
(1−α−L
)φ−c,
wh
ich
imp
liesφ≥
n−1
2−
c 2α
m-a
ssim
ilat
ion
As
abov
e,φ≥
n−1
2−
c 2α
e.g.
from
(m,A
)to
(MA,m
A).
φMu
tom
ult
icu
ltura
lism
M-a
ssim
ilat
ion
(1−L
)φ≥
(1−L
)(n−φ−
1)−c,
e.g.
from
(M,A
)to
(MA,m
B).
φMs
tose
greg
atio
nw
hic
him
pli
esφ≥
n−1
2−
c2(1−L)
m-a
ssim
ilat
ion
As
abov
e,φ≥
n−1
2−
c2(1−L)
e.g.
from
(m,A
)to
(MB,m
A).
φMs
tose
greg
atio
n
mu
ltic
ult
ura
lism
to(1−L
)(nM
+φ
)+
(1−α−L
)(nm−φ−
1)−c≥
e.g.
from
(MA,m
A)
to(M
,A).
M-a
ssim
ilat
ion
(1−L
)(nm−φ−
1)+
(1−α−L
)(nM
+φ
),φuM
wh
ich
imp
liesφ≥
nm−nM−1
2+
c 2α
[Th
ealt
ern
ati
veli
nk,
say
(MA,m
A)
to(M
,B),
has
wei
ght
at
least
nM−1
2+
1−α−L
2(1−L)nm
.]
mu
ltic
ult
ura
lism
to(1−L
)(nm
+φ
)+
(1−α−L
)(nM−φ−
1)−c≥
e.g.
from
(MA,m
A)
to(m,A
).φum
m-a
ssim
ilat
ion
(1−L
)(nM−φ−
1)+
(1−α−L
)(nm
+φ
),
wh
ich
imp
liesφ≥
nM−nm−1
2+
c 2α
[Th
ealt
enati
veli
nk,
say
(MA,m
A)
to(M
,B),
has
wei
ght
at
least
nm−1
2+
1−α−L
2(1−L)nM
.]
segr
egat
ion
to(1−L
)(nM
+φ
)−c≥
(1−L
)(nm−φ−
1),
e.g.
from
(MA,m
B)
to(M
,A).
φsM
M-a
ssim
ilat
ion
wh
ich
impli
esφ≥
nm−nM−1
2+
c2(1−L)
segr
egat
ion
to(1−L
)(nm
+φ
)−c≥
(1−L
)(nM−φ−
1),
e.g.
from
(MA,m
B)
to(m,B
).φsm
m-a
ssim
ilat
ion
wh
ich
impli
esφ≥
nM−nm−1
2+
c2(1−L)
44
Tab
leB
1co
nti
nu
ed
M-a
ssim
ilat
ion
(1−L
)φ+
(1−α−L
)(n−φ−
1)≥
e.g.
from
(M,A
)to
(m,A
).T
his
ism
inim
um
of
tom
-ass
imil
atio
n(1−L
)(n−φ−
1)+
(1−α−L
)φ−c;
an
dR
equ
ireφMu
mis
take
sfo
rm
tosw
itch
firs
t.m
ax{φ
Mu,φum}
(1−L
)(nm
+φ′ )
+(1−α−L
)(nM−φ′−
1)−c≥
Th
enM
wil
lsw
itch
ifφum
mis
take
sare
mad
e.
(1−L
)(nM−φ′−
1)+
(1−α−L
)(nm
+φ′ )
.an
d
(1−L
)φ≥
(1−L
)(n−φ−
1)−c;
an
dA
lter
nati
vely
exam
ine
e.g.
(M,A
)to
(m,B
).m
ax{φ
Ms,φsm}.
(1−L
)(nm
+φ′ )−c≥
(1−L
)(nM−φ′−
1).
Req
uir
eφMs
mis
take
sfo
rm
tosw
itch
firs
t.
Th
enM
swit
chifφsm
erro
rsare
mad
e.
m-a
ssim
ilat
ion
(1−L
)φ+
(1−α−L
)(n−φ−
1)≥
e.g.
from
(m,A
)to
(M,A
).T
his
ism
inim
um
of
toM
-ass
imil
atio
n(1−L
)(n−φ−
1)+
(1−α−L
)φ−c;
an
dφMu
mis
take
sre
qu
ired
forM
tosw
itch
firs
t.m
ax{φ
Mu,φuM}
(1−L
)(nM
+φ′ )
+(1−α−L
)(nm−φ′−
1)−c≥
Th
enm
wil
lsw
itch
ifφuM
erro
rsare
mad
e.
(1−L
)(nm−φ′−
1)+
(1−α−L
)(nM
+φ′ )
.an
d
(1−L
)φ≥
(1−L
)(n−φ−
1)−c;
an
dA
lter
nati
vely
exam
ine
e.g.
(m,A
)to
(M,B
).m
ax{φ
Ms,φsM}.
(1−L
)(nM
+φ′ )−c≥
(1−L
)(nm−φ′−
1).
φMs
mis
take
sre
qu
ired
forM
tosw
itch
firs
t.
Th
enm
wil
lsw
itch
ifφsM
erro
rsare
mad
e.
segr
egat
ion
to(1−L
)φ+
(1−α−L
)nM≥
(1−L
)(nm−φ−
1),
e.g.
from
(MA,m
B)
to(M
A,m
A).
mu
ltic
ult
ura
lism
wh
ich
imp
liesφ≥
nm−1
2−
1−α−L
2(1−L)nM
;an
d
(1−L
)φ+
(1−α−L
)nM≥
(1−L
)nM
+(1−α−L
)φ−c
Th
ese
con
din
equ
ali
tyen
sure
sm
do
not
wh
ich
impli
esφ≥nM−
c α.
want
top
lay
(M,A
).
(1−L
)φ+
(1−α−L
)nm≥
(1−L
)(nM−φ−
1),
Alt
ern
ati
vely
exam
ine
e.g.
from
(MA,m
B)
to(M
B,m
B).
wh
ich
imp
liesφ≥
nM−1
2−
1−α−L
2(1−L)nm
Ob
serv
eth
atM
do
not
want
top
lay
(m,B
).
mu
ltic
ult
ura
lism
(1−L
)φ≥
(1−L
)(nm−φ−
1)+
(1−α−L
)nM
,e.
g.
from
(MA,m
A)
to(M
A,m
B).
tose
greg
atio
nw
hic
him
pliesφ≥
nm−1
2+
1−α−L
2(1−L)nM
;an
d
(1−L
)φ≥
(1−L
)nM
+(1−α−L
)(nm−k−
1)−c,
Th
ese
con
din
equ
ali
tyen
sure
sm
do
not
wh
ich
impli
esφ≥
1−L
2(1−L)−αnM
+1−α−L
2(1−L)−α
(nm−
1)−
c2(1−L)−α
.w
ant
top
lay
(M,A
).
(1−L
)φ≥
(1−L
)(nM−φ−
1)+
(1−α−L
)nm
,A
lter
nati
vely
exam
ine
e.g.
from
(MA,m
A)
to(M
B,m
A).
wh
ich
impli
esφ≥
nM−1
2+
1−α−L
1−Lnm
;an
d
(1−L
)φ≥
(1−L
)nm
+(1−α−L
)(nM−k−
1),
Th
ese
con
din
equ
ali
tyen
sure
sM
do
not
wh
ich
imp
liesφ≥
1−L
2(1−L)−αnm
+1−α−L
2(1−L)−α
(nM−
1).
want
top
lay
(m,A
).
Commen
ts:
Wh
enα>L
forφMu<φuM
,th
ew
eight
for
alin
kfr
om
segre
gati
on
tom
ult
icu
ltu
ralism
islo
wer
than
spec
ified
ab
ove.
Th
isd
oes
not
chan
ge
the
resu
lts.
Mu
ltic
ult
ura
lism
tose
gre
gati
on
transi
tsco
nti
nu
eto
requ
ire
at
least
nm
−1
2+
1−α−L
2(1
−L)nM
mis
takes
.
45
1−α−L2(1−L)
nM mistakes. It can be seen from Table B1 that any link out of multiculturalism is of
weight at least the minimum of φuM and nm−12
+ 1−α−L2(1−L)
nM .
M-assimilation has lower stochastic potential than m-assimilation when φuM < φMu and φsm >
φMs.
Observe φum > φsm > φMs > φMu > φuM . Since φMu > φuM then nM > c/α and so φMs > φsMbut we know φMu < φMs and so φMu is the minimum weight transition from m-assimilation to
M -assimilation.
Without loss of generality consider the tree to (m,A). Suppose the tree has a link from
(M,A) to (m,A), which is of weight φum, or from (M,A) to (m,B), which is of weight φsm.
Form a link from (m,A) to (M,A), of weight φMu, and delete the link from (M,A) that goes to
either (m,A) or (m,B), to form a tree to (M,A) with strictly lower total weight. Suppose the
new tree has a link from (M,B) to (m,A), which is of weight φsm, or from (M,B) to (m,B),
which is of weight φum. Delete this link and form (m,A) to (M,B), of weight φMs, to form a
tree to (M,B) with strictly lower total weight.
Suppose instead the tree to (m,A) has no direct links from any M -assimilation node to any
m-assimilation node. Then there must be a direct link from segregation and/or multiculturalism
to some m-assimilation node. Suppose the direct link is from (MA;mA) to either (m,A) or
(m,B). This link is of weight at least φum or nm−12
+ 1−α−L2(1−L)
nM . Delete such a link and from
the same node send the link instead to (M,A) of weight φuM < nm−12
.62 Now form a link from
(m,A) to (M,A) of weight φMu and delete any link from (M,A) which are all of weight at least
φMu. Suppose the direct link is from (MA;mB) to either (m,A) or (m,B). This link is of
weight at least φsm. Delete such a link and from the same node send the link instead to (M,A)
of weight φsM . Now form a link from (m,A) to (M,A) of weight φMu and delete any link from
(M,A) which are all of weight at least φMu.
Finally suppose none of the other links are present in the tree to (m,A). Then there must
be a link from (mB;mA) or (MB;mB) to either (m,A) or (m,B) or both, and no other direct
links. Transfer all such links to (M,B) which, as above, produces a graph of strictly lower total
weight. Every node now has a path to (M,B) apart from (m,A) (and possibly (m,B) if it links
directly to (m,A)). So finally form a link from (m,A) to (MA;mA) of weight φMu and delete
a link from (M,B) which is at least φMu.
For φ ∈ R, under any parameters there is a unique set, Σx, with lowest stochastic potential.
However, in a finite population φ is discrete. Because of this discreteness we can get intervals
where the set is not unique. To avoid issues associated with discreteness we examine the param-
eter range |φuM−φMu| ≥ 1. Then φuM−φMu ≥ 1 implies nm ≥ n−c/α+1, and φMu−φuM ≥ 1
implies nm ≤ n−c/α−1. To avoid the same issues associated with discreteness close to parame-
ter values 1−L = α we also require nm−12
+ 1−α−L2(1−L)
nM−min{φMu, φuM} ≥ 1. Since φuM +φMu =
nm− 1 combined with the assumption φuM −φMu ≥ 1 implies min{φMu, φuM} ≤ nm−12− 1
2, this
implies 1− L ≥ α + 1nM−1
.63 Finally Assumption 2 continues to ensure φsm − φMs ≥ 1.
62Observe that the difference between φuM and φum is the same as the difference between φsM and φsm.63This also guarantees that the alternative link from segregation to mulitculturalism has weight at least one unit higher than the
alternative link from multiculturalism to segregation: nm−12
+ 1−α−L2(1−L)
nM − [nm−12− 1−α−L
2(1−L)nM ] ≥ 1.
46
Online Appendix C Additional empirical results
Table C1: Number of observed cohorts by nationality
Minimum cohort size of
(1) 30 (2) 20 (3) 40
Germany 303 455 234
Italy 291 404 225
Poland 196 276 149
Canada 174 259 116
Britain 167 248 127
Russia 151 219 122
Sweden 139 242 96
Ireland 133 185 109
Mexico 59 102 38
Other 133 227 91
Total 1746 2617 1307
Notes. Number of cohorts we observe, broken down by nationality. Cohorts are defined by nationality, county, tenure in the US
grouped to the nearest 10 years, and year of arrival in 10 year bands. Column (1) shows the number of cohorts when we require
cohorts to contain a minimum of 30 people. Columns (2) and (3) show how these results change when thresholds of 20 or 40 are
used instead. Germany includes Germans and Austrians. Sweden includes Sweden, Denmark, and Norway. Other is composed (in
order of size, at cohort size 30) of Romania, Japan, Netherlands, Greece, Finland, Portugal, West Indies, France, China, Spain,
Belgium, Cuba, South America. These nationalities are treated as seperate from each other in our analysis, and are aggregated
here only for brevity.
47
Tab
leC
2:C
omp
arin
gth
eth
resh
old
effec
tin
share
spea
kin
gE
nglish
,u
sin
gd
iffer
ent
sam
ple
rest
rict
ion
sor
wei
ghts
Dep
end
ent
Var
iab
le:
Pro
por
tion
ofp
eop
lein
the
coh
ort
that
spea
kE
ngli
sh
Sta
nd
ard
Err
ors
inP
aren
thes
es
Coh
orts
defi
ned
by
Nat
ion
alit
y,C
ounty
,G
rou
ped
Yea
rof
Arr
ival,
Gro
up
edT
enure
Min
imu
mco
hort
size
(1)
30
(2)
20
(3)
40
(4)
Wei
ghte
dre
gre
ssio
n
Op
tim
alth
resh
old
(β1|τ
=τ∗ )
-.50
9***
-.348***
-.475***
-.510***
(.025)
(.016)
(.024)
(.035)
Con
stan
t(β
0)
.885
***
.885***
.890***
.886***
(.004)
(.004)
(.004)
(.004)
Op
tim
alT
hre
shol
dL
evel
(τ∗ )
.31
.27
.30
.31
F-s
tati
stic
422
457
394
208
1%cr
itic
alva
lue
for
F-s
tati
stic
6.6
6.6
6.6
6.6
Coh
ort
Fix
edE
ffec
tsN
oN
oN
oN
o
Slo
pes
inN
atio
nal
ity
Sh
are
No
No
No
No
Ob
serv
atio
ns
1272
1925
955
1272
Notes.
***
den
ote
ssi
gn
ifica
nce
at
0.1
%,
**
at
1%
,an
d*
at
5%
level
,w
hen
trea
ted
as
ast
an
dalo
ne
regre
ssio
n.
Th
eou
tcom
em
easu
res
the
pro
port
ion
of
the
coh
ort
that
spea
ks
En
gli
sh.
Coh
ort
s
wit
hE
nglish
,C
an
ad
ian
,an
dIr
ish
nati
on
aliti
esare
excl
ud
edfr
om
the
sam
ple
,si
nce
En
glish
islikel
yto
be
thei
rm
oth
erto
ngu
e.C
oh
ort
fixed
effec
tsare
com
pose
dof
nati
on
ality
,arr
ival
yea
r
(gro
up
ed),
an
dte
nu
re(g
rou
ped
)fi
xed
effec
ts.
All
colu
mn
sare
regre
ssio
ns
of
share
spea
kin
gE
nglish
on
ase
qu
ence
of
nati
on
ality
share
thre
shold
s.W
evary
the
thre
shold
bet
wee
n.2
0an
d.4
0,
at
inte
rvals
of
.01.
We
pro
vid
eth
ere
sult
sfo
rth
e
thre
shold
am
on
gth
ese
wh
ich
pro
du
ced
are
gre
ssio
nw
ith
the
hig
hes
tF
-sta
tist
icw
hen
test
edagain
stth
enu
llof
no
thre
shold
.T
he
valu
efo
rth
isth
resh
old
,th
ees
tim
ate
dF
-sta
tist
ic,
an
dth
e1%
crit
ical
valu
efo
rth
isst
ati
stic
(An
dre
ws,
1993)
are
pro
vid
edat
the
bott
om
of
the
tab
le.
Colu
mn
(1)
requ
ires
am
inim
um
coh
ort
size
of
30,
as
inour
main
spec
ifica
tion
s.C
olu
mn
s(2
)an
d(3
)u
sem
inim
um
size
sof
20
an
d40
resp
ecti
vel
y.C
olu
mn
(4)
reta
ins
the
min
imu
mco
hort
size
of
30,
bu
tfi
rst
div
ides
coh
ort
sin
the
un
rest
rict
edsa
mp
lein
tod
eciles
base
don
nati
on
ality
share
,an
dth
enw
eights
ob
serv
ati
on
sin
each
of
thes
ed
eciles
inth
ean
aly
sis
sam
ple
inver
sely
pro
port
ion
ally
toth
esi
zeof
that
dec
ile
inth
ean
aly
sis
sam
ple
.
48
Tab
leC
3:C
omp
arin
gth
eth
resh
old
effec
tin
share
in-m
arr
ied
,u
sin
gd
iffer
ent
sam
ple
rest
rict
ion
sor
wei
ghts
Dep
end
ent
Var
iab
le:
Pro
por
tion
ofm
arri
edp
eop
lein
the
coh
ort
that
are
marr
ied
toso
meo
ne
of
the
sam
en
ati
on
ali
ty
Sta
nd
ard
Err
ors
inP
aren
thes
es
Coh
orts
defi
ned
by
Nat
ion
alit
y,C
ounty
,G
rou
ped
Yea
rof
Arr
ival,
Gro
up
edT
enure
Min
imu
mco
hort
size
(1)
30
(2)
20
(3)
40
(4)
Wei
ghte
dre
gre
ssio
n
Op
tim
alth
resh
old
(β1|τ
=τ∗ )
.151
***
.094**
.128**
.142***
(.042)
(.030)
(.048)
(.030)
Con
stan
t(β
0)
.606
***
.598***
.613***
.609***
(.006)
(.005)
(.006)
(.006)
Op
tim
alT
hre
shol
dL
evel
(τ∗ )
.35
.35
.35
.33
F-s
tati
stic
13.0
9.7
7.3
22.4
1%cr
itic
alva
lue
for
F-s
tati
stic
6.6
6.6
6.6
6.6
Coh
ort
Fix
edE
ffec
tsN
oN
oN
oN
o
Slo
pes
inN
atio
nal
ity
Sh
are
No
No
No
No
Ob
serv
atio
ns
1746
2617
1307
1746
Notes.
***
den
ote
ssi
gn
ifica
nce
at
0.1
%,
**
at
1%
,an
d*
at
5%
level
,w
hen
trea
ted
as
ast
and
alo
ne
regre
ssio
n.
Th
eou
tcom
em
easu
res
the
marr
ied
pro
port
ion
of
the
coh
ort
that
is‘in
-marr
ied
’i.e.
marr
ied
toso
meo
ne
of
the
sam
en
ati
on
ality
.C
oh
ort
fixed
effec
tsare
com
pose
dof
nati
on
ality
,arr
ival
yea
r(g
rou
ped
),an
dte
nu
re(g
rou
ped
)fi
xed
effec
ts.
All
colu
mn
sare
regre
ssio
ns
of
share
in-m
arr
ied
on
ase
qu
ence
of
nati
on
ality
share
thre
shold
s.W
evary
the
thre
shold
bet
wee
n.2
0an
d.4
0,
at
inte
rvals
of
.01.
We
pro
vid
eth
ere
sult
sfo
rth
eth
resh
old
am
on
gth
ese
wh
ich
pro
du
ced
are
gre
ssio
nw
ith
the
hig
hes
tF
-sta
tist
icw
hen
test
edagain
stth
enu
llof
no
thre
shold
.T
he
valu
efo
rth
isth
resh
old
,th
ees
tim
ate
dF
-sta
tist
ic,
an
dth
e1%
crit
ical
valu
e
for
this
stati
stic
(An
dre
ws,
1993)
are
pro
vid
edat
the
bott
om
of
the
tab
le.
Colu
mn
(1)
requ
ires
am
inim
um
coh
ort
size
of
30,
as
inou
rm
ain
spec
ifica
tion
s.C
olu
mn
s(2
)an
d(3
)u
sem
inim
um
size
sof
20
an
d40
resp
ecti
vel
y.C
olu
mn
(4)
reta
ins
the
min
imu
mco
hort
size
of
30,
bu
tfirs
td
ivid
esco
hort
sin
the
un
rest
rict
edsa
mp
lein
tod
eciles
base
don
nati
on
ality
share
,an
dth
enw
eights
ob
serv
ati
on
sin
each
of
thes
ed
eciles
inth
ean
aly
sis
sam
ple
inver
sely
pro
port
ion
ally
toth
esi
zeof
that
dec
ile
inth
ean
aly
sis
sam
ple
.
49
